Transposed Poisson structures on generalized Witt algebras and Block Lie algebras

Abstract

We describe transposed Poisson structures on generalized Witt algebras $W(A,V,⟨\cdot ,\cdot ⟩)$ and Block Lie algebras $L(A,g,f)$ over a field $F$ of characteristic zero, where $⟨\cdot ,\cdot ⟩$ and $f$ are non-degenerate. More specifically, if $\dim (V)>1$, then all the transposed Poisson algebra structures on $W(A,V,⟨\cdot ,\cdot ⟩)$ are trivial; and if $\dim (V)=1$, then such structures are, up to isomorphism, mutations of the group algebra structure on $FA$. The transposed Poisson algebra structures on $L(A,g,f)$ are in a one-to-one correspondence with commutative and associative multiplications defined on a complement of the square of $L(A,g,f)$ with values in the center of $L(A,g,f)$. In particular, all of them are usual Poisson structures on $L(A,g,f)$. This generalizes earlier results about transposed Poisson structures on Block Lie algebras $\mathcal{B}(q)$.

References in corpus (1)

• The algebraic and geometric classification of transposed Poisson algebras

Cited by in corpus (4)

• Non-associative algebraic structures: classification and structure
• Transposed Poisson structures on solvable and perfect Lie algebras
• Transposed Poisson structures on the Lie algebra of upper triangular matrices
• Transposed Poisson structures on Lie incidence algebras