In abstract algebra, a Valya algebra (or Valentina algebra) is a nonassociative algebra M over a field F whose multiplicative binary operation g satisfies the following axioms:

1. The skew-symmetry condition

for all .

2. The Valya identity

for all , where k=1,2,...,6, and

3. The bilinear condition

for all and .

We say that M is a Valya algebra if the commutant of this algebra is a Lie subalgebra. Each Lie algebra is a Valya algebra.

There is the following relationship between the commutant-associative algebra and Valentina algebra. The replacement of the multiplication g(A,B) in an algebra M by the operation of commutation [A,B]=g(A,B)-g(B,A), makes it into the algebra . If M is a commutant-associative algebra, then is a Valya algebra. A Valya algebra is a generalization of a Lie algebra.

Examples

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Let us give the following examples regarding Valya algebras.

(1) Every finite Valya algebra is the tangent algebra of an analytic local commutant-associative loop (Valya loop) as each finite Lie algebra is the tangent algebra of an analytic local group (Lie group). This is the analog of the classical correspondence between analytic local groups (Lie groups) and Lie algebras.

(2) A bilinear operation for the differential 1-forms

 

on a symplectic manifold can be introduced by the rule

 

where   is 1-form. A set of all nonclosed 1-forms, together with this operation, is Lie algebra.

If   and   are closed 1-forms, then   and

 

A set of all closed 1-forms, together with this bracket, form a Lie algebra. A set of all nonclosed 1-forms together with the bilinear operation   is a Valya algebra, and it is not a Lie algebra.

See also

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References

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  • A. Elduque, H. C. Myung Mutations of alternative algebras, Kluwer Academic Publishers, Boston, 1994, ISBN 0-7923-2735-7
  • V.T. Filippov (2001) [1994], "Mal'tsev algebra", Encyclopedia of Mathematics, EMS Press
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  • A.G. Kurosh, General algebra. Lectures for the academic year 1969/70. Nauka, Moscow,1974. (In Russian)
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  • 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
  • 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.
  • Zhevlakov, K.A. (2001) [1994], "Alternative rings and algebras", Encyclopedia of Mathematics, EMS Press