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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'', ''B''] = ''AB'' − ''BA'' is the [[commutator]] of ''A'' and ''B'' and | |||
(''A'', ''B'', ''C'') = (''AB'')''C'' – ''A''(''BC'') is the [[associator]] of ''A'', ''B'' and ''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'', ''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 | 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 [http://books.google.ru/books?id=pHK11tfdE3QC&dq=V.E.+Tarasov+Quantum+Mechanics+of+Non-Hamiltonian+and+Dissipative+Systems.&printsec=frontcover&source=bl&ots=qDERzjAJd9&sig=U8V7RUVd1SW8mx4GzE1T-2canhA&hl=ru&ei=pkvkSeycINiEsAbloKSfCw&sa=X&oi=book_result&ct=result&resnum=1 ''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]] | |||
Revision as of 04:03, 24 October 2013
In abstract algebra, a commutant-associative algebra is a nonassociative algebra over a field whose multiplication satisfies the following axiom:
where [A, B] = AB − BA is the commutator of A and B and (A, B, C) = (AB)C – A(BC) is the associator of A, B and C.
In other words, an algebra M is commutant-associative if the commutant, i.e. the subalgebra of M generated by all commutators [A, B], is an associative algebra.
See also
References
- A. Elduque, H. C. Myung Mutations of alternative algebras, Kluwer Academic Publishers, Boston, 1994, ISBN 0-7923-2735-7
- Other Sports Official Kull from Drumheller, has hobbies such as telescopes, property developers in singapore and crocheting. Identified some interesting places having spent 4 months at Saloum Delta.
my web-site http://himerka.com/ - M.V. Karasev, V.P. Maslov, Nonlinear Poisson Brackets: Geometry and Quantization. American Mathematical Society, Providence, 1993.
- 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
- A.G. Kurosh, General algebra. Lectures for the academic year 1969/70. Nauka, Moscow,1974. (In Russian)
- A.I. Mal'tsev, Algebraic systems. Springer, 1973. (Translated from Russian)
- A.I. Mal'tsev, Analytic loops. Mat. Sb., 36 : 3 (1955) pp. 569–576 (In Russian)
- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 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 Quantum Mechanics of Non-Hamiltonian and Dissipative Systems. Elsevier Science, Amsterdam, Boston, London, New York, 2008. ISBN 0-444-53091-6 ISBN 9780444530912
- Template:Eom