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 12-derivation of I(X,K) decomposes into the sum of a central-valued 12-derivation, an inner 12-derivation and a 12-derivation associated with a map σ:X<2→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 λ:Xe2→K, where Xe2 is the set of (x,y)∈X2 such that x<y is a maximal chain not contained in a cycle.