Depolarization and distributive laws

Abstract

Given a vector space with two multiplications, one commutative the other anticommutative, possibly connected by a distributive law, the depolarization principle allows to look at this triplet through a single nonassociative multiplication. This is the case of Poisson algebras. We are interested here in the cases of transposed Poisson algebras and we show in this case that depolarization cannot be done with a single multiplication. We also examine the depolarization for Hom-Lie algebras.