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

${\displaystyle g(A,B)=-g(B,A)}$

for all ${\displaystyle A,B\in M}$.

2. The Valya identity

${\displaystyle J(g(A_{1},A_{2}),g(A_{3},A_{4}),g(A_{5},A_{6}))=0}$

for all ${\displaystyle A_{k}\in M}$, where k=1,2,...,6, and

${\displaystyle J(A,B,C):=g(g(A,B),C)+g(g(B,C),A)+g(g(C,A),B).}$

3. The bilinear condition

${\displaystyle g(aA+bB,C)=ag(A,C)+bg(B,C)}$

for all ${\displaystyle A,B,C\in M}$ and ${\displaystyle a,b\in F}$.

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 ${\displaystyle M^{(-)}}$. If M is a commutant-associative algebra, then ${\displaystyle M^{(-)}}$ is a Valya algebra. A Valya algebra is a generalization of a Lie algebra.

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

${\displaystyle \alpha =F_{k}(x)\,dx^{k},\quad \beta =G_{k}(x)\,dx^{k}}$

on a symplectic manifold can be introduced by the rule

${\displaystyle (\alpha ,\beta )_{0}=d\Psi (\alpha ,\beta )+\Psi (d\alpha ,\beta )+\Psi (\alpha ,d\beta ),\,}$

where ${\displaystyle (\alpha ,\beta )}$ is 1-form. A set of all nonclosed 1-forms, together with this operation, is Lie algebra.

If ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ are closed 1-forms, then ${\displaystyle d\alpha =d\beta =0}$ and

${\displaystyle (\alpha ,\beta )=d\Psi (\alpha ,\beta ).\,}$

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 ${\displaystyle (\alpha ,\beta )}$ is a Valya algebra, and it is not a Lie algebra.

• Malcev algebra
• Alternative algebra
• Commutant-associative algebra

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