Transposed Poisson structures on generalized Witt algebras and Block Lie algebras

TL;DR

This paper classifies transposed Poisson structures on generalized Witt and Block Lie algebras over a field of characteristic zero, revealing triviality in higher dimensions and a correspondence with associative multiplications in certain cases.

Contribution

It provides a complete description of transposed Poisson structures on these classes of Lie algebras, extending previous results and clarifying their structure in various dimensional settings.

Findings

All transposed Poisson structures are trivial when the dimension of V exceeds 1.

Structures are mutations of the group algebra when dim(V)=1.

On Block Lie algebras, structures correspond to associative multiplications on specific subspaces.

Abstract

We describe transposed Poisson structures on generalized Witt algebras $W(A,V, \langle \cdot,\cdot \rangle )$ and Block Lie algebras $L(A,g,f)$ over a field $F$ of characteristic zero, where $\langle \cdot,\cdot \rangle$ and $f$ are non-degenerate. More specifically, if $\dim(V)>1$, then all the transposed Poisson algebra structures on $W(A,V,\langle \cdot,\cdot \rangle)$ 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 $\mathcal{B}(q)$.