On simple transposed Poisson algebras

TL;DR

This paper develops a structure theory for transposed Poisson algebras, classifies simple finite-dimensional cases over algebraically closed fields, and explores their representations and applications.

Contribution

It introduces a decomposition theorem for finite-dimensional transposed Poisson algebras and classifies simple cases over algebraically closed fields of characteristic p>3.

Findings

Every finite-dimensional transposed Poisson algebra decomposes into a unital and a nilpotent ideal.

Simple finite-dimensional transposed Poisson algebras have underlying Lie algebra as a Zassenhaus algebra.

The irreducible finite-dimensional representations of these algebras are determined.

Abstract

We develop a structure theory for transposed Poisson algebras over fields of characteristic different from two. In particular, we prove that every finite-dimensional transposed Poisson algebra over an algebraically closed field decomposes as the direct sum of a unital ideal and a nilpotent ideal. As a consequence, we obtain restrictions on simple transposed Poisson algebras and use them to classify the simple finite-dimensional transposed Poisson algebras over an algebraically closed field of characteristic $p>3$. Precisely, we show that every such algebra has as underlying Lie algebra a Zassenhaus algebra $\mathcal{W}(1;n)$ and is isomorphic to one of the algebras of the family $\mathcal{W}_n(q)$ arising from a mutation of a natural associative commutative structure on $\mathcal{W}(1;n)$. We then study the corresponding isomorphism problem for the family $\mathcal{W}_n(q)$ and determine the irreducible finite-dimensional representations of these simple transposed Poisson algebras $\mathcal{W}_n(q)$ in the unital case. We conclude with some applications to Jordan superalgebras, weak-Leibniz algebras and quasi-Poisson algebras.