Transposed Poisson structures on Lie incidence algebras

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 1/2-derivation of I(X, K) decomposes into the sum of a central-valued 1/2-derivation, an inner 1/2-derivation and a 1/2-derivation associated with a map sigma : 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 lambda : X-is an element of(2) -> K , where X(is an element of)(2 )is the set of (x, y) is an element of X-2 such that x < y is a maximal chain not contained in a cycle. (c) 2024 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons .org /licenses /by /4 .0/).