Poisson Derivations and the First Poisson Cohomology Group on Trivial Extension Algebras

Abstract

Let A be a Poisson algebra and M be a left Poisson module. We construct a Poisson structure on the trivial extension algebra $$A\ltimes M$$ . We investigate, in detail, Poisson derivations and Hamiltonian derivations on $$A\ltimes M$$ . As a consequence, we characterize the first Poisson cohomology group of $$A\ltimes M$$ in terms of the ones of A and M. In the case of that A is finite dimensional and $$M=A^*$$ , we show that $${\text {HP}}^1(A)$$ is a summand of $${\text {HP}}^1(A\ltimes M)$$ . These are generalization of the results on the derivation and the first Hochschild cohomology group of trivial extension algebras in Assem et al. (J Pure Appl Algebra 220:2471–2499, 2016), Bennis and Fahid (Medieter J Math 14:150, https://doi.org/10.1007/s00009-017-0949-z , 2017), Cibils et al. (Glasg Math J 45:21–40, 2003) to Poisson framework.