Lie-admissible structures on Witt type algebras

Abstract

International audienceIn this paper we study Lie-admissible structures on Witt type algebras. Witt type algebras are Γ -graded Lie algebras (where Γ is an abelian group) which generalize the Witt algebra. We give all third power-associative and flexible Lie-admissible structures on these algebras. In particular we generalize some results on the Witt algebra. After describing the second scalar cohomology group of Witt type algebras, we investigate third power- associative and flexible Lie-admissible structures on the central extension of some Witt type algebras. Finally we study a left-symmetric structure induced by a symplectic formfor some Witt type algebras

Transposed Poisson structures on Block Lie algebras and superalgebras

“…In what follows, we assume ( 4) and (5). It is also natural to assume that |f (Γ)| ≥ 2, since otherwise V (f ) is abelian.…”

“…They are playing a critical role in the classification of simple Lie algebras on a lattice [10]. Lie admissible structures on Witt type algebras have been described in [5].…”

Weakly associative algebras, Poisson algebras and quantization

“…the sums being on all permutations of (1,2,3,4). This expression has to satisfy δ 3 W A • δ 2 W A = 0.…”