Non-associative algebras | Lie algebras

Valya algebra

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. (Wikipedia).

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