Valya algebra


In abstract algebra, a Valya 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 in an algebra M by the operation of commutation =g-g, 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

Let us give the following examples regarding Valya algebras.
Every finite Valya algebra is the tangent algebra of an analytic local commutant-associative loop as each finite Lie algebra is the tangent algebra of an analytic local group. This is the analog of the classical correspondence between analytic local groups and Lie algebras.
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.