View source for Commutant-associative algebra

In [[abstract algebra]], a '''commutant-associative algebra''' is a [[Algebra over a field#Non-associative algebras|nonassociative algebra over a field]] whose [[product (mathematics)|multiplication]] satisfies the following axiom:

:<math> ([A_1,A_2], [A_3,A_4], [A_5,A_6]) =0 </math>,

where [''A'',&nbsp;''B'']&nbsp;=&nbsp;''AB''&nbsp;−&nbsp;''BA'' is the [[commutator]] of ''A'' and ''B'' and
(''A'',&nbsp;''B'',&nbsp;''C'') = (''AB'')''C''&nbsp;–&nbsp;''A''(''BC'') is the [[associator]] of ''A'', ''B'' and&nbsp;''C''.

In other words, an algebra ''M'' is commutant-associative if the commutant, i.e. the subalgebra of ''M'' generated by all [[commutator]]s [''A'',&nbsp;''B''], is an [[Associativity|associative]] algebra.

==See also==
* [[Valya algebra]]
* [[Malcev algebra]]
* [[Alternative algebra]]

==References==
* A. Elduque,  H. C. Myung ''Mutations of alternative algebras'',  Kluwer Academic Publishers, Boston, 1994, {{ISBN|0-7923-2735-7}}
* {{springer|id=M/m062170|author=V.T. Filippov|title=Mal'tsev algebra}}
* M.V. Karasev, V.P. Maslov, ''Nonlinear Poisson Brackets: Geometry and Quantization.'' American Mathematical Society, Providence, 1993.
* [[Aleksandr Gennadievich Kurosh|A.G. Kurosh]], ''Lectures on general algebra.'' Translated from the Russian edition (Moscow, 1960) by K. A. Hirsch. Chelsea, New York, 1963. 335 pp. {{ISBN|0-8284-0168-3}}  {{ISBN|978-0-8284-0168-5}} 
* [[Aleksandr Gennadievich Kurosh|A.G. Kurosh]], ''General algebra. Lectures for the academic year 1969/70.'' Nauka, Moscow,1974.  (In Russian)
* [[Anatoly Maltsev|A.I. Mal'tsev]], ''Algebraic systems.'' Springer, 1973.  (Translated from Russian)
* [[Anatoly Maltsev|A.I. Mal'tsev]], '' Analytic loops.''  Mat. Sb., 36 : 3  (1955)  pp. 569–576  (In Russian)
*{{cite book | first = R.D. | last = Schafer | title = An Introduction to Nonassociative Algebras | url = https://archive.org/details/introductiontono0000scha | url-access = registration | publisher = Dover Publications | location = New York | year = 1995 | isbn = 0-486-68813-5}}
* [http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=tmf&paperid=962&option_lang=eng V.E. Tarasov, "Quantum dissipative systems: IV. Analogues of Lie algebras and groups" Theoretical and Mathematical Physics. Vol.110. No.2. (1997) pp.168-178.] 
* V.E. Tarasov [https://books.google.com/books?id=pHK11tfdE3QC&q=V.E.+Tarasov+Quantum+Mechanics+of+Non-Hamiltonian+and+Dissipative+Systems. ''Quantum Mechanics of Non-Hamiltonian and Dissipative Systems.'' Elsevier Science, Amsterdam, Boston, London, New York, 2008.] {{ISBN|0-444-53091-6}} {{ISBN|9780444530912}}
*{{eom|id=A/a012090|first=K.A.|last= Zhevlakov|title=Alternative rings and algebras}}

[[Category:Non-associative algebras]]