Transposed Poisson structures on Lie incidence algebras

Abstract

Let $X$ be a finite connected poset, $K$ a field of characteristic zero and $I(X,K)$ the incidence algebra of $X$ over $K$ seen as a Lie algebra under the commutator product. In the first part of the paper we show that any $\frac{1}{2}$-derivation of $I(X,K)$ decomposes into the sum of a central-valued $\frac 12$-derivation, an inner $\frac{1}{2}$-derivation and a $\frac{1}{2}$-derivation associated with a map $\sigma:X^2_<\to K$ that is constant on chains and cycles in $X$. In the second part of the paper we use this result to prove that any transposed Poisson structure on $I(X,K)$ is the sum of a structure of Poisson type, a mutational structure and a structure determined by $\lambda:X^2_e\to K$, where $X^2_e$ is the set of $(x,y)\in X^2$ such that $x<y$ is a maximal chain not contained in a cycle.