Commutant-associative algebra

In abstract algebra, a commutant-associative algebra is a nonassociative algebra over a field whose multiplication satisfies the following axiom:

[math]\displaystyle{ ([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 commutators [A, B], is an 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
* Hazewinkel, Michiel, ed. (2001), "Mal'tsev algebra", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=M/m062170
* 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)
* Schafer, R.D. (1995). An Introduction to Nonassociative Algebras. New York: Dover Publications. ISBN 0-486-68813-5. https://archive.org/details/introductiontono0000scha.
* 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
* Hazewinkel, Michiel, ed. (2001), "Alternative rings and algebras", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=A/a012090

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