Universal enveloping algebra

In mathematics, the universal enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra.

Universal enveloping algebras are used in the representation theory of Lie groups and Lie algebras. For example, Verma modules can be constructed as quotients of the universal enveloping algebra.^[1] In addition, the enveloping algebra gives a precise definition for the Casimir operators. Because Casimir operators commute with all elements of a Lie algebra, they can be used to classify representations. The precise definition also allows the importation of Casimir operators into other areas of mathematics, specifically, those that have a differential algebra. They also play a central role in some recent developments in mathematics. In particular, their dual provides a commutative example of the objects studied in non-commutative geometry, the quantum groups. This dual can be shown, by the Gelfand–Naimark theorem, to contain the C* algebra of the corresponding Lie group. This relationship generalizes to the idea of Tannaka–Krein duality between compact topological groups and their representations.

From an analytic viewpoint, the universal enveloping algebra of the Lie algebra of a Lie group may be identified with the algebra of left-invariant differential operators on the group.

Informal construction

The idea of the universal enveloping algebra is to embed a Lie algebra Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{g}} into an associative algebra Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{A}} with identity in such a way that the abstract bracket operation in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{g}} corresponds to the commutator Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle xy-yx} in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{A}} and the algebra Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{A}} is generated by the elements of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{g}} . There may be many ways to make such an embedding, but there is a unique "largest" such Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{A}} , called the universal enveloping algebra of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{g}} .

Generators and relations

Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{g}} be a Lie algebra, assumed finite-dimensional for simplicity, with basis Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle X_{1},\ldots X_{n}} . Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{ijk}} be the structure constants for this basis, so that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [X_i,X_j]=\sum_{k=1}^n c_{ijk}X_k.}

Then the universal enveloping algebra is the associative algebra (with identity) generated by elements Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_1,\ldots x_n} subject to the relations

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_i x_j - x_j x_i=\sum_{k=1}^n c_{ijk}x_k}

and no other relations. Below we will make this "generators and relations" construction more precise by constructing the universal enveloping algebra as a quotient of the tensor algebra over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak g} .

Consider, for example, the Lie algebra sl(2,C), spanned by the matrices

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E = \begin{pmatrix} 0 & 1\\ 0 & 0 \end{pmatrix} \qquad F = \begin{pmatrix} 0 & 0\\ 1 & 0 \end{pmatrix} \qquad H = \begin{pmatrix} 1 & 0\\ 0 & -1 \end{pmatrix} ~,}

which satisfy the commutation relations Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [H,E]=2E} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [H,F]=-2F} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [E,F]=H} . The universal enveloping algebra of sl(2,C) is then the algebra generated by three elements Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e,f,h} subject to the relations

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle he-eh=2e,\quad hf-fh=-2f,\quad ef-fe=h,}

and no other relations. We emphasize that the universal enveloping algebra is not the same as (or contained in) the algebra of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2\times 2} matrices. For example, the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2\times 2} matrix Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E} satisfies Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E^2=0} , as is easily verified. But in the universal enveloping algebra, the element Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} does not satisfy Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^2=0} because we do not impose this relation in the construction of the enveloping algebra. Indeed, it follows from the Poincaré–Birkhoff–Witt theorem (discussed § below) that the elements Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1,e,e^2,e^3,\ldots} are all linearly independent in the universal enveloping algebra.

Finding a basis

In general, elements of the universal enveloping algebra are linear combinations of products of the generators in all possible orders. Using the defining relations of the universal enveloping algebra, we can always re-order those products in a particular order, say with all the factors of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_1} first, then factors of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_2} , etc. For example, whenever we have a term that contains Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_2 x_1} (in the "wrong" order), we can use the relations to rewrite this as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_1 x_2} plus a linear combination of the Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x_{j}} 's. Doing this sort of thing repeatedly eventually converts any element into a linear combination of terms in ascending order. Thus, elements of the form

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_1^{k_1}x_2^{k_2}\cdots x_n^{k_n}}

with the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_j} 's being non-negative integers, span the enveloping algebra. (We allow Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_j=0} , meaning that we allow terms in which no factors of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_j} occur.) The Poincaré–Birkhoff–Witt theorem, discussed below, asserts that these elements are linearly independent and thus form a basis for the universal enveloping algebra. In particular, the universal enveloping algebra is always infinite dimensional.

The Poincaré–Birkhoff–Witt theorem implies, in particular, that the elements Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_1,\ldots, x_n} themselves are linearly independent. It is therefore common—if potentially confusing—to identify the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_j} 's with the generators Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_j} of the original Lie algebra. That is to say, we identify the original Lie algebra as the subspace of its universal enveloping algebra spanned by the generators. Although Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{g}} may be an algebra of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n\times n} matrices, the universal enveloping of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{g}} does not consist of (finite-dimensional) matrices. In particular, there is no finite-dimensional algebra that contains the universal enveloping of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{g}} ; the universal enveloping algebra is always infinite dimensional. Thus, in the case of sl(2,C), if we identify our Lie algebra as a subspace of its universal enveloping algebra, we must not interpret Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} and $H$ as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2\times 2} matrices, but rather as symbols with no further properties (other than the commutation relations).

Formalities

The formal construction of the universal enveloping algebra takes the above ideas, and wraps them in notation and terminology that makes it more convenient to work with. The most important difference is that the free associative algebra used in the above is narrowed to the tensor algebra, so that the product of symbols is understood to be the tensor product. The commutation relations are imposed by constructing a quotient space of the tensor algebra quotiented by the smallest two-sided ideal containing elements of the form Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_i x_j -x_j x_i-\Sigma c_{ijk}x_k} . The universal enveloping algebra is the "largest" unital associative algebra generated by elements of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak g} with a Lie bracket compatible with the original Lie algebra.

Formal definition

Recall that every Lie algebra Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{g}} is in particular a vector space. Thus, one is free to construct the tensor algebra Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T(\mathfrak{g})} from it. The tensor algebra is a free algebra: it simply contains all possible tensor products of all possible vectors in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{g}} , without any restrictions whatsoever on those products.

That is, one constructs the space

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T(\mathfrak{g}) = K \,\oplus\, \mathfrak{g} \,\oplus\, (\mathfrak{g} \otimes \mathfrak{g}) \,\oplus\, (\mathfrak{g} \otimes \mathfrak{g} \otimes \mathfrak{g}) \,\oplus\, \cdots }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \otimes} is the tensor product, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \oplus} is the direct sum of vector spaces. Here, $K$ is the field over which the Lie algebra is defined. From here, through to the remainder of this article, the tensor product is always explicitly shown. Many authors omit it, since, with practice, its location can usually be inferred from context. Here, a very explicit approach is adopted, to minimize any possible confusion about the meanings of expressions.

The first step in the construction is to "lift" the Lie bracket from the Lie algebra (where it is defined) to the tensor algebra (where it is not), so that one can coherently work with the Lie bracket of two tensors. The lifting is done as follows. First, recall that the bracket operation on a Lie algebra is a map Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{g}\times\mathfrak{g}\to\mathfrak{g}} that is bilinear, skew-symmetric and satisfies the Jacobi identity. We wish to define a Lie bracket [-,-] that is a map Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T(\mathfrak{g})\otimes T(\mathfrak{g})\to T(\mathfrak{g})} that is also bilinear, skew symmetric and obeys the Jacobi identity.

The lifting can be done grade by grade. Begin by defining the bracket on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{g} \otimes \mathfrak{g} \to \mathfrak{g}} as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \otimes b - b \otimes a = [a,b]}

This is a consistent, coherent definition, because both sides are bilinear, and both sides are skew symmetric (the Jacobi identity will follow shortly). The above defines the bracket on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T^2(\mathfrak{g})=\mathfrak{g} \otimes \mathfrak{g}} ; it must now be lifted to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T^n(\mathfrak{g})} for arbitrary Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n.} This is done recursively, by defining

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [a\otimes b, c] = a \otimes [b,c] + [a,c]\otimes b}

and likewise

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [a, b\otimes c] = [a,b]\otimes c + b\otimes [a,c]}

It is straightforward to verify that the above definition is bilinear, and is skew-symmetric; one can also show that it obeys the Jacobi identity. The final result is that one has a Lie bracket that is consistently defined on all of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T(\mathfrak{g});} one says that it has been "lifted" to all of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T(\mathfrak{g})} in the conventional sense of a "lift" from a base space (here, the Lie algebra) to a covering space (here, the tensor algebra).

The result of this lifting is explicitly a Poisson algebra. It is a unital associative algebra with a Lie bracket that is compatible with the Lie algebra bracket; it is compatible by construction. It is not the smallest such algebra, however; it contains far more elements than needed. One can get something smaller by projecting back down. The universal enveloping algebra Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U(\mathfrak{g})} of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{g}} is defined as the quotient space

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U(\mathfrak{g}) = T(\mathfrak{g})/\sim}

where the equivalence relation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sim} is given by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a\otimes b - b \otimes a = [a,b]}

That is, the Lie bracket defines the equivalence relation used to perform the quotienting. The result is still a unital associative algebra, and one can still take the Lie bracket of any two members. Computing the result is straight-forward, if one keeps in mind that each element of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U(\mathfrak{g})} can be understood as a coset: one just takes the bracket as usual, and searches for the coset that contains the result. It is the smallest such algebra; one cannot find anything smaller that still obeys the axioms of an associative algebra.

The universal enveloping algebra is what remains of the tensor algebra after modding out the Poisson algebra structure. (This is a non-trivial statement; the tensor algebra has a rather complicated structure: it is, among other things, a Hopf algebra; the Poisson algebra is likewise rather complicated, with many peculiar properties. It is compatible with the tensor algebra, and so the modding can be performed. The Hopf algebra structure is conserved; this is what leads to its many novel applications, e.g. in string theory. However, for the purposes of the formal definition, none of this particularly matters.)

The construction can be performed in a slightly different (but ultimately equivalent) way. Forget, for a moment, the above lifting, and instead consider the two-sided ideal $I$ generated by elements of the form

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a\otimes b - b \otimes a - [a,b]}

This generator is an element of

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{g} \oplus (\mathfrak{g}\otimes\mathfrak{g}) \subset T(\mathfrak{g})}

A general member of the ideal $I$ will have the form

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c\otimes d \otimes \cdots \otimes (a\otimes b - b \otimes a - [a,b]) \otimes f \otimes g \cdots}

for some Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a,b,c,d,f,g\in\mathfrak{g}.} All elements of $I$ are obtained as linear combinations of elements of this form. Clearly, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I\subset T(\mathfrak{g})} is a subspace. It is an ideal, in that if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j\in I} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\in T(\mathfrak{g}),} then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j\otimes x\in I} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\otimes j\in I.} Establishing that this is an ideal is important, because ideals are precisely those things that one can quotient with; ideals lie in the kernel of the quotienting map. That is, one has the short exact sequence

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0\to I \to T(\mathfrak{g}) \to T(\mathfrak{g})/I \to 0}

where each arrow is a linear map, and the kernel of that map is given by the image of the previous map. The universal enveloping algebra can then be defined as^[2]

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U(\mathfrak{g}) = T(\mathfrak{g})/I}

Superalgebras and other generalizations

The above construction focuses on Lie algebras and on the Lie bracket, and its skewness and antisymmetry. To some degree, these properties are incidental to the construction. Consider instead some (arbitrary) algebra (not a Lie algebra) over a vector space, that is, a vector space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} endowed with multiplication Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m:V\times V\to V} that takes elements Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a\times b\mapsto m(a,b).} If the multiplication is bilinear, then the same construction and definitions can go through. One starts by lifting $m$ up to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T(V)} so that the lifted Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} obeys all of the same properties that the base Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} does – symmetry or antisymmetry or whatever. The lifting is done exactly as before, starting with

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} m: V \otimes V &\to V \\ a \otimes b &\mapsto m(a,b) \end{align}}

This is consistent precisely because the tensor product is bilinear, and the multiplication is bilinear. The rest of the lift is performed so as to preserve multiplication as a homomorphism. By definition, one writes

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m(a \otimes b,c)= a \otimes m(b,c) + m(a,c) \otimes b}

and also that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m(a,b\otimes c)= m(a,b) \otimes c + b \otimes m(a,c)}

This extension is consistent by appeal to a lemma on free objects: since the tensor algebra is a free algebra, any homomorphism on its generating set can be extended to the entire algebra. Everything else proceeds as described above: upon completion, one has a unital associative algebra; one can take a quotient in either of the two ways described above.

The above is exactly how the universal enveloping algebra for Lie superalgebras is constructed. One need only to carefully keep track of the sign, when permuting elements. In this case, the (anti-)commutator of the superalgebra lifts to an (anti-)commuting Poisson bracket.

Another possibility is to use something other than the tensor algebra as the covering algebra. One such possibility is to use the exterior algebra; that is, to replace every occurrence of the tensor product by the exterior product. If the base algebra is a Lie algebra, then the result is the Gerstenhaber algebra; it is the exterior algebra of the corresponding Lie group. As before, it has a grading naturally coming from the grading on the exterior algebra. (The Gerstenhaber algebra should not be confused with the Poisson superalgebra; both invoke anticommutation, but in different ways.)

The construction has also been generalized for Malcev algebras,^[3] Bol algebras^[4] and left alternative algebras.^{[citation needed]}

Universal property

The universal enveloping algebra, or rather the universal enveloping algebra together with the canonical map Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h\colon\mathfrak{g}\to U(\mathfrak{g})} , possesses a universal property.^[5] Suppose we have any Lie algebra map

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi: \mathfrak{g} \to A}

to a unital associative algebra $A$ (with Lie bracket in $A$ given by the commutator). More explicitly, this means that we assume

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi([X,Y])=\varphi(X)\varphi(Y)-\varphi(Y)\varphi(X)}

for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X,Y\in\mathfrak{g}} . Then there exists a unique unital algebra homomorphism

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \widehat\varphi\colon U(\mathfrak{g}) \to A}

such that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi = \widehat \varphi \circ h }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h:\mathfrak{g}\to U(\mathfrak{g})} is the canonical map. (The map Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h} is obtained by embedding Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{g}} into its tensor algebra and then composing with the quotient map to the universal enveloping algebra. This map is an embedding, by the Poincaré–Birkhoff–Witt theorem.)

To put it differently, if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi\colon\mathfrak{g}\rightarrow A} is a linear map into a unital algebra Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} satisfying Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi([X,Y])=\varphi(X)\varphi(Y)-\varphi(Y)\varphi(X)} , then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi} extends to an algebra homomorphism of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \widehat\varphi: U(\mathfrak{g}) \to A} . Since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U(\mathfrak{g})} is generated by elements of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{g}} , the map ${\widehat {\varphi }}$ must be uniquely determined by the requirement that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \widehat{\varphi}(X_{i_1}\cdots X_{i_N})=\varphi(X_{i_1})\cdots \varphi(X_{i_N}),\quad X_{i_j}\in\mathfrak{g}} .

The point is that because there are no other relations in the universal enveloping algebra besides those coming from the commutation relations of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{g}} , the map Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \widehat{\varphi}} is well defined, independent of how one writes a given element Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\in U(\mathfrak{g})} as a linear combination of products of Lie algebra elements.

The universal property of the enveloping algebra immediately implies that every representation of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{g}} acting on a vector space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} extends uniquely to a representation of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U(\mathfrak{g})} . (Take Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A=\mathrm{End}(V)} .) This observation is important because it allows (as discussed below) the Casimir elements to act on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} . These operators (from the center of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U(\mathfrak{g})} ) act as scalars and provide important information about the representations. The quadratic Casimir element is of particular importance in this regard.

Other algebras

Although the canonical construction, given above, can be applied to other algebras, the result, in general, does not have the universal property. Thus, for example, when the construction is applied to Jordan algebras, the resulting enveloping algebra contains the special Jordan algebras, but not the exceptional ones: that is, it does not envelope the Albert algebras. Likewise, the Poincaré–Birkhoff–Witt theorem, below, constructs a basis for an enveloping algebra; it just won't be universal. Similar remarks hold for the Lie superalgebras.

Poincaré–Birkhoff–Witt theorem

The Poincaré–Birkhoff–Witt theorem gives a precise description of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U(\mathfrak{g})} . This can be done in either one of two different ways: either by reference to an explicit vector basis on the Lie algebra, or in a coordinate-free fashion.

Using basis elements

One way is to suppose that the Lie algebra can be given a totally ordered basis, that is, it is the free vector space of a totally ordered set. Recall that a free vector space is defined as the space of all finitely supported functions from a set $X$ to the field $K$ (finitely supported means that only finitely many values are non-zero); it can be given a basis Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e_a:X\to K} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e_a(b) = \delta_{ab}} is the indicator function for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a,b\in X} . Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h:\mathfrak{g}\to T(\mathfrak{g})} be the injection into the tensor algebra; this is used to give the tensor algebra a basis as well. This is done by lifting: given some arbitrary sequence of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e_a} , one defines the extension of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h} to be

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h(e_a\otimes e_b \otimes\cdots \otimes e_c) = h(e_a) \otimes h(e_b) \otimes\cdots \otimes h(e_c)}

The Poincaré–Birkhoff–Witt theorem then states that one can obtain a basis for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U(\mathfrak{g})} from the above, by enforcing the total order of $X$ onto the algebra. That is, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U(\mathfrak{g})} has a basis

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e_a\otimes e_b \otimes\cdots \otimes e_c}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a\le b \le \cdots \le c} , the ordering being that of total order on the set $X$ .^[6] The proof of the theorem involves noting that, if one starts with out-of-order basis elements, these can always be swapped by using the commutator (together with the structure constants). The hard part of the proof is establishing that the final result is unique and independent of the order in which the swaps were performed.

This basis should be easily recognized as the basis of a symmetric algebra. That is, the underlying vector spaces of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U(\mathfrak{g})} and the symmetric algebra are isomorphic, and it is the PBW theorem that shows that this is so. See, however, the section on the algebra of symbols, below, for a more precise statement of the nature of the isomorphism.

It is useful, perhaps, to split the process into two steps. In the first step, one constructs the free Lie algebra: this is what one gets, if one mods out by all commutators, without specifying what the values of the commutators are. The second step is to apply the specific commutation relations from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{g}.} The first step is universal, and does not depend on the specific Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{g}.} It can also be precisely defined: the basis elements are given by Hall words, a special case of which are the Lyndon words; these are explicitly constructed to behave appropriately as commutators.

Coordinate-free

One can also state the theorem in a coordinate-free fashion, avoiding the use of total orders and basis elements. This is convenient when there are difficulties in defining the basis vectors, as there can be for infinite-dimensional Lie algebras. It also gives a more natural form that is more easily extended to other kinds of algebras. This is accomplished by constructing a filtration Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_m \mathfrak{g}} whose limit is the universal enveloping algebra Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U(\mathfrak{g}).}

First, a notation is needed for an ascending sequence of subspaces of the tensor algebra. Let

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_m\mathfrak{g} = K\oplus \mathfrak{g}\oplus T^2\mathfrak{g} \oplus \cdots \oplus T^m\mathfrak{g}}

where

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T^m\mathfrak{g} = T^{\otimes m} \mathfrak{g} = \mathfrak{g}\otimes \cdots \otimes \mathfrak{g}}

is the $m$ -times tensor product of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{g}.} The Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_m\mathfrak{g}} form a filtration:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K\subset \mathfrak{g}\subset T_2\mathfrak{g} \subset \cdots \subset T_m\mathfrak{g} \subset\cdots}

More precisely, this is a filtered algebra, since the filtration preserves the algebraic properties of the subspaces. Note that the limit of this filtration is the tensor algebra Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T(\mathfrak{g}).}

It was already established, above, that quotienting by the ideal is a natural transformation that takes one from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T(\mathfrak{g})} to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U(\mathfrak{g}).} This also works naturally on the subspaces, and so one obtains a filtration Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_m \mathfrak{g}} whose limit is the universal enveloping algebra Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U(\mathfrak{g}).}

Next, define the space

G_{m}{\mathfrak {g}}=U_{m}{\mathfrak {g}}/U_{m-1}{\mathfrak {g}}

This is the space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_m \mathfrak{g}} modulo all of the subspaces Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_n \mathfrak{g}} of strictly smaller filtration degree. Note that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G_m\mathfrak{g}} is not at all the same as the leading term Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U^m\mathfrak{g}} of the filtration, as one might naively surmise. It is not constructed through a set subtraction mechanism associated with the filtration.

Quotienting Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_m \mathfrak{g}} by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_{m-1} \mathfrak{g}} has the effect of setting all Lie commutators defined in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_m \mathfrak{g}} to zero. One can see this by observing that the commutator of a pair of elements whose products lie in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_{m} \mathfrak{g}} actually gives an element in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_{m-1} \mathfrak{g}} . This is perhaps not immediately obvious: to get this result, one must repeatedly apply the commutation relations, and turn the crank. The essence of the Poincaré–Birkhoff–Witt theorem is that it is always possible to do this, and that the result is unique.

Since commutators of elements whose products are defined in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_{m} \mathfrak{g}} lie in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_{m-1} \mathfrak{g}} , the quotienting that defines Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G_m\mathfrak{g}} has the effect of setting all commutators to zero. What PBW states is that the commutator of elements in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G_m\mathfrak{g}} is necessarily zero. What is left are the elements that are not expressible as commutators.

In this way, one is lead immediately to the symmetric algebra. This is the algebra where all commutators vanish. It can be defined as a filtration Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_m \mathfrak{g}} of symmetric tensor products Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{Sym}^m \mathfrak{g}} . Its limit is the symmetric algebra Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S(\mathfrak{g})} . It is constructed by appeal to the same notion of naturality as before. One starts with the same tensor algebra, and just uses a different ideal, the ideal that makes all elements commute:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S(\mathfrak{g}) = T(\mathfrak{g}) / (a\otimes b - b\otimes a)}

Thus, one can view the Poincaré–Birkhoff–Witt theorem as stating that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G(\mathfrak{g})} is isomorphic to the symmetric algebra Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S(\mathfrak{g})} , both as a vector space and as a commutative algebra.

The Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G_m\mathfrak{g}} also form a filtered algebra; its limit is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G(\mathfrak{g}).} This is the associated graded algebra of the filtration.

The construction above, due to its use of quotienting, implies that the limit of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G(\mathfrak{g})} is isomorphic to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U(\mathfrak{g}).} In more general settings, with loosened conditions, one finds that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S(\mathfrak{g})\to G(\mathfrak{g})} is a projection, and one then gets PBW-type theorems for the associated graded algebra of a filtered algebra. To emphasize this, the notation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{gr}U(\mathfrak{g})} is sometimes used for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G(\mathfrak{g}),} serving to remind that it is the filtered algebra.

Other algebras

The theorem, applied to Jordan algebras, yields the exterior algebra, rather than the symmetric algebra. In essence, the construction zeros out the anti-commutators. The resulting algebra is an enveloping algebra, but is not universal. As mentioned above, it fails to envelop the exceptional Jordan algebras.

Left-invariant differential operators

Suppose Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} is a real Lie group with Lie algebra Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{g}} . Following the modern approach, we may identify Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{g}} with the space of left-invariant vector fields (i.e., first-order left-invariant differential operators). Specifically, if we initially think of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{g}} as the tangent space to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} at the identity, then each vector in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{g}} has a unique left-invariant extension. We then identify the vector in the tangent space with the associated left-invariant vector field. Now, the commutator (as differential operators) of two left-invariant vector fields is again a vector field and again left-invariant. We can then define the bracket operation on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{g}} as the commutator on the associated left-invariant vector fields.^[7] This definition agrees with any other standard definition of the bracket structure on the Lie algebra of a Lie group.

We may then consider left-invariant differential operators of arbitrary order. Every such operator Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} can be expressed (non-uniquely) as a linear combination of products of left-invariant vector fields. The collection of all left-invariant differential operators on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} forms an algebra, denoted Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D(G)} . It can be shown that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D(G)} is isomorphic to the universal enveloping algebra Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U(\mathfrak{g})} .^[8]

In the case that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{g}} arises as the Lie algebra of a real Lie group, one can use left-invariant differential operators to give an analytic proof of the Poincaré–Birkhoff–Witt theorem. Specifically, the algebra Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D(G)} of left-invariant differential operators is generated by elements (the left-invariant vector fields) that satisfy the commutation relations of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{g}} . Thus, by the universal property of the enveloping algebra, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D(G)} is a quotient of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U(\mathfrak{g})} . Thus, if the PBW basis elements are linearly independent in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D(G)} —which one can establish analytically—they must certainly be linearly independent in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U(\mathfrak{g})} . (And, at this point, the isomorphism of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D(G)} with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U(\mathfrak{g})} is apparent.)

Algebra of symbols

The underlying vector space of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S(\mathfrak{g})} may be given a new algebra structure so that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U(\mathfrak{g})} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S(\mathfrak{g})} are isomorphic as associative algebras. This leads to the concept of the algebra of symbols Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \star(\mathfrak{g})} : the space of symmetric polynomials, endowed with a product, the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \star} , that places the algebraic structure of the Lie algebra onto what is otherwise a standard associative algebra. That is, what the PBW theorem obscures (the commutation relations) the algebra of symbols restores into the spotlight.

The algebra is obtained by taking elements of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S(\mathfrak{g})} and replacing each generator Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e_i} by an indeterminate, commuting variable Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t_i} to obtain the space of symmetric polynomials Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K[t_i]} over the field Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K} . Indeed, the correspondence is trivial: one simply substitutes the symbol Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t_i} for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e_i} . The resulting polynomial is called the symbol of the corresponding element of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S(\mathfrak{g})} . The inverse map is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w: \star(\mathfrak{g})\to U(\mathfrak{g})}

that replaces each symbol Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t_i} by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e_i} . The algebraic structure is obtained by requiring that the product Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \star} act as an isomorphism, that is, so that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w(p \star q) = w(p)\otimes w(q)}

for polynomials Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p,q\in \star(\mathfrak{g}).}

The primary issue with this construction is that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w(p)\otimes w(q)} is not trivially, inherently a member of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U(\mathfrak{g})} , as written, and that one must first perform a tedious reshuffling of the basis elements (applying the structure constants as needed) to obtain an element of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U(\mathfrak{g})} in the properly ordered basis. An explicit expression for this product can be given: this is the Berezin formula.^[9] It follows essentially from the Baker–Campbell–Hausdorff formula for the product of two elements of a Lie group.

A closed form expression is given by^[10]

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(t)\star q(t)= \left. \exp\left(t_i m^i \left(\frac{\partial}{\partial u}, \frac{\partial}{\partial v} \right) \right) p(u)q(v)\right \vert_{u=v=t}}

where

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m(A,B)=\log\left(e^Ae^B\right)-A-B}

and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m^i} is just Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} in the chosen basis.

The universal enveloping algebra of the Heisenberg algebra is the Weyl algebra (modulo the relation that the center be the unit); here, the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \star} product is called the Moyal product.

Representation theory

The universal enveloping algebra preserves the representation theory: the representations of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{g}} correspond in a one-to-one manner to the modules over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U(\mathfrak{g})} . In more abstract terms, the abelian category of all representations of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{g}} is isomorphic to the abelian category of all left modules over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U(\mathfrak{g})} .

The representation theory of semisimple Lie algebras rests on the observation that there is an isomorphism, known as the Kronecker product:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U(\mathfrak{g}_1\oplus\mathfrak{g}_2)\cong U(\mathfrak{g}_1)\otimes U(\mathfrak{g}_2)}

for Lie algebras Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{g}_1, \mathfrak{g}_2} . The isomorphism follows from a lifting of the embedding

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i(\mathfrak{g}_1 \oplus \mathfrak{g}_2) =i_1(\mathfrak{g}_1)\otimes 1 \oplus 1\otimes i_2(\mathfrak{g}_2)}

where

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i:\mathfrak{g}\to U(\mathfrak{g})}

is just the canonical embedding (with subscripts, respectively for algebras one and two). It is straightforward to verify that this embedding lifts, given the prescription above. See, however, the discussion of the bialgebra structure in the article on tensor algebras for a review of some of the finer points of doing so: in particular, the shuffle product employed there corresponds to the Wigner-Racah coefficients, i.e. the 6j and 9j-symbols, etc.

Also important is that the universal enveloping algebra of a free Lie algebra is isomorphic to the free associative algebra.

Construction of representations typically proceeds by building the Verma modules of the highest weights.

In a typical context where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{g}} is acting by infinitesimal transformations, the elements of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U(\mathfrak{g})} act like differential operators, of all orders. (See, for example, the realization of the universal enveloping algebra as left-invariant differential operators on the associated group, as discussed above.)

Casimir operators

The center Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z(U(\mathfrak{g}))} of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U(\mathfrak{g})} can be identified with the centralizer of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{g}} in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U(\mathfrak{g}).} Any element of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z(U(\mathfrak{g}))} must commute with all of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U(\mathfrak{g}),} and in particular with the canonical embedding of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{g}} into Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U({\mathfrak {g}}).} Because of this, the center is directly useful for classifying representations of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{g}} . For a finite-dimensional semisimple Lie algebra, the Casimir operators form a distinguished basis from the center Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z(U(\mathfrak{g}))} . These may be constructed as follows.

The center Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z(U(\mathfrak{g}))} corresponds to linear combinations of all elements Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z=v\otimes w \otimes \cdots \otimes u \in U(\mathfrak{g})} that commute with all elements Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\in \mathfrak{g};} that is, for which Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [z,x]=\mbox{ad}_x(z)=0.} That is, they are in the kernel of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mbox{ad}_\mathfrak{g}.} Thus, a technique is needed for computing that kernel. What we have is the action of the adjoint representation on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{g};} we need it on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U(\mathfrak{g}).} The easiest route is to note that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mbox{ad}_\mathfrak{g}} is a derivation, and that the space of derivations can be lifted to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T(\mathfrak{g})} and thus to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U(\mathfrak{g}).} This implies that both of these are differential algebras.

By definition, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta:\mathfrak{g}\to\mathfrak{g}} is a derivation on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{g}} if it obeys Leibniz's law:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta([v,w])=[\delta(v),w]+[v,\delta(w)]}

(When Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{g}} is the space of left invariant vector fields on a group Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} , the Lie bracket is that of vector fields.) The lifting is performed by defining

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align}\delta(v\otimes w \otimes \cdots \otimes u) =& \, \delta(v) \otimes w \otimes \cdots \otimes u \\ &+ v\otimes \delta(w) \otimes \cdots\otimes u \\ &+ \cdots + v\otimes w \otimes \cdots \otimes \delta(u). \end{align} }

Since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mbox{ad}_x} is a derivation for any Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\in\mathfrak{g},} the above defines Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mbox{ad}_x} acting on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T(\mathfrak{g})} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U(\mathfrak{g}).}

From the PBW theorem, it is clear that all central elements are linear combinations of symmetric homogenous polynomials in the basis elements Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e_a} of the Lie algebra. The Casimir invariants are the irreducible homogenous polynomials of a given, fixed degree. That is, given a basis Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e_a} , a Casimir operator of order Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} has the form

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_{(m)} = \kappa^{ab\cdots c}e_a\otimes e_b\otimes \cdots\otimes e_c}

where there are Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} terms in the tensor product, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \kappa^{ab\cdots c}} is a completely symmetric tensor of order Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} belonging to the adjoint representation. That is, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \kappa^{ab\cdots c}} can be (should be) thought of as an element of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(\operatorname{ad}_\mathfrak{g}\right)^{\otimes m}.} Recall that the adjoint representation is given directly by the structure constants, and so an explicit indexed form of the above equations can be given, in terms of the Lie algebra basis; this is originally a theorem of Israel Gel'fand. That is, from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [x,C_{(m)}]=0} , it follows that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_{ij}^{\;\; k} \kappa^{jl\cdots m} + f_{ij}^{\;\; l} \kappa^{kj\cdots m} + \cdots + f_{ij}^{\;\; m} \kappa^{kl\cdots j} = 0 }

where the structure constants are

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [e_i,e_j]=f_{ij}^{\;\; k}e_k}

As an example, the quadratic Casimir operator is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_{(2)} = \kappa^{ij} e_i\otimes e_j}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \kappa^{ij}} is the inverse matrix of the Killing form Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \kappa_{ij}.} That the Casimir operator Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_{(2)}} belongs to the center Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z(U(\mathfrak{g}))} follows from the fact that the Killing form is invariant under the adjoint action.

The center of the universal enveloping algebra of a simple Lie algebra is given in detail by the Harish-Chandra isomorphism.

Rank

The number of algebraically independent Casimir operators of a finite-dimensional semisimple Lie algebra is equal to the rank of that algebra, i.e. is equal to the rank of the Cartan–Weyl basis. This may be seen as follows. For a $d$ -dimensional vector space $V$ , recall that the determinant is the completely antisymmetric tensor on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V^{\otimes d}} . Given a matrix $M$ , one may write the characteristic polynomial of $M$ as

\det(tI-M)=\sum _{n=0}^{d}p_{n}t^{n}

For a $d$ -dimensional Lie algebra, that is, an algebra whose adjoint representation is $d$ -dimensional, the linear operator

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \operatorname {ad} :{\mathfrak {g}}\to \operatorname {End} ({\mathfrak {g}})}

implies that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{ad}_x} is a $d$ -dimensional endomorphism, and so one has the characteristic equation

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \det(tI-\operatorname{ad}_x)=\sum_{n=0}^d p_n(x)t^n}

for elements Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\in \mathfrak{g}.} The non-zero roots of this characteristic polynomial (that are roots for all $x$ ) form the root system of the algebra. In general, there are only $r$ such roots; this is the rank of the algebra. This implies that the highest value of $n$ for which the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_n(x)} is non-vanishing is $r .$

The Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_n(x)} are homogeneous polynomials of degree $d - n .$ This can be seen in several ways: Given a constant Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k\in K} , ad is linear, so that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{ad}_{kx}=k\,\operatorname{ad}_x.} By plugging and chugging in the above, one obtains that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_n(kx)=k^{d-n}p_n(x).}

By linearity, if one expands in the basis,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=\sum_{i=1}^d x_i e_i}

then the polynomial has the form

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_n(x)=x_ax_b\cdots x_c \kappa^{ab\cdots c}}

that is, a Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \kappa} is a tensor of rank Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m=d-n} . By linearity and the commutativity of addition, i.e. that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{ad}_{x+y}=\operatorname{ad}_{y+x},} , one concludes that this tensor must be completely symmetric. This tensor is exactly the Casimir invariant of order $m .$

The center Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z(\mathfrak{g})} corresponded to those elements Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z\in Z(\mathfrak{g})} for which Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{ad}_x(z)=0} for all $x;$ by the above, these clearly corresponds to the roots of the characteristic equation. One concludes that the roots form a space of rank $r$ and that the Casimir invariants span this space. That is, the Casimir invariants generate the center Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z(U(\mathfrak{g})).}

Example: Rotation group SO(3)

The rotation group SO(3) is of rank one, and thus has one Casimir operator. It is three-dimensional, and thus the Casimir operator must have order (3 − 1) = 2 i.e. be quadratic. Of course, this is the Lie algebra of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_1.} As an elementary exercise, one can compute this directly. Changing notation to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e_i=L_i,} with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_i} belonging to the adjoint rep, a general algebra element is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle xL_1+yL_2+zL_3} and direct computation gives

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \det\left(xL_1+yL_2+zL_3-tI\right)=-t^3-(x^2+y^2+z^2)t}

The quadratic term can be read off as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \kappa^{ij}=\delta^{ij}} , and so the squared angular momentum operator for the rotation group is that Casimir operator. That is,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_{(2)} = L^2 = e_1\otimes e_1 + e_2\otimes e_2 + e_3\otimes e_3}

and explicit computation shows that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [L^2, e_k]=0}

after making use of the structure constants

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [e_i, e_j]=\varepsilon_{ij}^{\;\;k}e_k}

Example: Pseudo-differential operators

A key observation during the construction of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U(\mathfrak{g})} above was that it was a differential algebra, by dint of the fact that any derivation on the Lie algebra can be lifted to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U(\mathfrak{g})} . Thus, one is led to a ring of pseudo-differential operators, from which one can construct Casimir invariants.

If the Lie algebra Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{g}} acts on a space of linear operators, such as in Fredholm theory, then one can construct Casimir invariants on the corresponding space of operators. The quadratic Casimir operator corresponds to an elliptic operator.

If the Lie algebra acts on a differentiable manifold, then each Casimir operator corresponds to a higher-order differential on the cotangent manifold, the second-order differential being the most common and most important.

If the action of the algebra is isometric, as would be the case for Riemannian or pseudo-Riemannian manifolds endowed with a metric and the symmetry groups SO(N) and SO (P, Q), respectively, one can then contract upper and lower indices (with the metric tensor) to obtain more interesting structures. For the quadratic Casimir invariant, this is the Laplacian. Quartic Casimir operators allow one to square the stress–energy tensor, giving rise to the Yang-Mills action. The Coleman–Mandula theorem restricts the form that these can take, when one considers ordinary Lie algebras. However, the Lie superalgebras are able to evade the premises of the Coleman–Mandula theorem, and can be used to mix together space and internal symmetries.

Examples in particular cases

If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{g} = \mathfrak{sl}_2} , then it has a basis of matrices

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H = \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}, \text{ } E = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \text{ } F = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}}

which satisfy the following identities under the standard bracket:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [H,E] = -2E} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [H,F] = 2F} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [E,F] = - H }

this shows us that the universal enveloping algebra has the presentation

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U(\mathfrak{sl}_2) = \frac{\mathbb{C}\langle x,y,z\rangle}{(xy - yx + 2y, xz - zx - 2z, yz - zy + x)}}

as a non-commutative ring.

If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{g}} is abelian (that is, the bracket is always $0$ ), then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U(\mathfrak{g})} is commutative; and if a basis of the vector space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{g}} has been chosen, then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U(\mathfrak{g})} can be identified with the polynomial algebra over $K$ , with one variable per basis element.

If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{g}} is the Lie algebra corresponding to the Lie group $G$ , then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U(\mathfrak{g})} can be identified with the algebra of left-invariant differential operators (of all orders) on $G$ ; with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{g}} lying inside it as the left-invariant vector fields as first-order differential operators.

To relate the above two cases: if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{g}} is a vector space $V$ as abelian Lie algebra, the left-invariant differential operators are the constant coefficient operators, which are indeed a polynomial algebra in the partial derivatives of first order.

The center Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z(\mathfrak{g})} consists of the left- and right- invariant differential operators; this, in the case of $G$ not commutative, is often not generated by first-order operators (see for example Casimir operator of a semi-simple Lie algebra).

Another characterization in Lie group theory is of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U(\mathfrak{g})} as the convolution algebra of distributions supported only at the identity element $e$ of $G$ .

The algebra of differential operators in $n$ variables with polynomial coefficients may be obtained starting with the Lie algebra of the Heisenberg group. See Weyl algebra for this; one must take a quotient, so that the central elements of the Lie algebra act as prescribed scalars.

The universal enveloping algebra of a finite-dimensional Lie algebra is a filtered quadratic algebra.

Hopf algebras and quantum groups

The construction of the group algebra for a given group is in many ways analogous to constructing the universal enveloping algebra for a given Lie algebra. Both constructions are universal and translate representation theory into module theory. Furthermore, both group algebras and universal enveloping algebras carry natural comultiplications that turn them into Hopf algebras. This is made precise in the article on the tensor algebra: the tensor algebra has a Hopf algebra structure on it, and because the Lie bracket is consistent with (obeys the consistency conditions for) that Hopf structure, it is inherited by the universal enveloping algebra.

Given a Lie group $G$ , one can construct the vector space $C(G)$ of continuous complex-valued functions on $G$ , and turn it into a C*-algebra. This algebra has a natural Hopf algebra structure: given two functions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi, \psi\in C(G)} , one defines multiplication as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\nabla(\varphi, \psi))(x)=\varphi(x)\psi(x)}

and comultiplication as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\Delta(\varphi))(x\otimes y)=\varphi(xy),}

the counit as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon(\varphi)=\varphi(e)}

and the antipode as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (S(\varphi))(x)=\varphi(x^{-1}).}

Now, the Gelfand–Naimark theorem essentially states that every commutative Hopf algebra is isomorphic to the Hopf algebra of continuous functions on some compact topological group $G$ —the theory of compact topological groups and the theory of commutative Hopf algebras are the same. For Lie groups, this implies that $C(G)$ is isomorphically dual to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U(\mathfrak{g})} ; more precisely, it is isomorphic to a subspace of the dual space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U^*(\mathfrak{g}).}

These ideas can then be extended to the non-commutative case. One starts by defining the quasi-triangular Hopf algebras, and then performing what is called a quantum deformation to obtain the quantum universal enveloping algebra, or quantum group, for short.

References

^ Hall 2015 Section 9.5
^ Hall 2015 Section 9.3
^ Perez-Izquierdo, J.M.; Shestakov, I.P. (2004). "An envelope for Malcev algebras". Journal of Algebra. 272: 379–393. doi:10.1016/s0021-8693(03)00389-2. hdl:10338.dmlcz/140108.
^ Perez-Izquierdo, J.M. (2005). "An envelope for Bol algebras". Journal of Algebra. 284 (2): 480–493. doi:10.1016/j.jalgebra.2004.09.038.
^ Hall 2015 Theorem 9.7
^ Hall 2015 Theorem 9.10
^ E.g. Helgason 2001 Chapter II, Section 1
^ Helgason 2001 Chapter II, Proposition 1.9
^ Berezin, F.A. (1967). "Some remarks about the associated envelope of a Lie algebra". Funct. Anal. Appl. 1 (2): 91. doi:10.1007/bf01076082. S2CID 122356554.
^ Xavier Bekaert, "Universal enveloping algebras and some applications in physics" (2005) Lecture, Modave Summer School in Mathematical Physics.

Dixmier, Jacques (1996) [1974], Enveloping algebras, Graduate Studies in Mathematics, vol. 11, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0560-2, MR 0498740
Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, ISBN 978-3319134666
Helgason, Sigurdur (2001), Differential geometry, Lie groups, and symmetric spaces, Graduate Studies in Mathematics, vol. 34, Providence, R.I.: American Mathematical Society, doi:10.1090/gsm/034, ISBN 978-0-8218-2848-9, MR 1834454, S2CID 120016227
Musson, Ian M. (2012), Lie Superalgebras and Enveloping Algebras, Graduate Studies in Mathematics, vol. 131, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-6867-6, Zbl 1255.17001
Shlomo Sternberg (2004), Lie algebras, Harvard University.
Universal enveloping algebra at the nLab

[1] Hall 2015 Section 9.5

[2] Hall 2015 Section 9.3

[3] Perez-Izquierdo, J.M.; Shestakov, I.P. (2004). "An envelope for Malcev algebras". Journal of Algebra. 272: 379–393. doi:10.1016/s0021-8693(03)00389-2. hdl:10338.dmlcz/140108.

[4] Perez-Izquierdo, J.M. (2005). "An envelope for Bol algebras". Journal of Algebra. 284 (2): 480–493. doi:10.1016/j.jalgebra.2004.09.038.

[5] Hall 2015 Theorem 9.7

[6] Hall 2015 Theorem 9.10

[7] E.g. Helgason 2001 Chapter II, Section 1

[8] Helgason 2001 Chapter II, Proposition 1.9

[9] Berezin, F.A. (1967). "Some remarks about the associated envelope of a Lie algebra". Funct. Anal. Appl. 1 (2): 91. doi:10.1007/bf01076082. S2CID 122356554.

[10] Xavier Bekaert, "Universal enveloping algebras and some applications in physics" (2005) Lecture, Modave Summer School in Mathematical Physics.

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Informal construction

Generators and relations

Finding a basis

Formalities

Formal definition

Superalgebras and other generalizations

Universal property

Other algebras

Poincaré–Birkhoff–Witt theorem

Using basis elements

Coordinate-free

Other algebras

Left-invariant differential operators

Algebra of symbols

Representation theory

Casimir operators

Rank

Example: Rotation group SO(3)

Example: Pseudo-differential operators

Examples in particular cases

Hopf algebras and quantum groups

See also

References