Valya Algebra
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High Quality Content by WIKIPEDIA articles! 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: 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 M( ). If M is ...
High Quality Content by WIKIPEDIA articles! 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: 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 M( ). If M is a commutant-associative algebra, then M( ) is a Valya algebra. A Valya algebra is a generalization of a Lie algebra. Let us give the following examples regarding Valya algebras. 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.
