Hochschild cohomology of relation extension algebras

Let B be the split extension of a finite dimensional algebra C by a C-C-bimodule E. We define a morphism of associative graded algebras phi* : HH*(B)) -> HH* (C) from the Hochschild cohomology of B to that of C, extending similar constructions for the first cohomology groups made and studied by Assem, Bustamante, Igusa, Redondo and Schiflier. In the case of a trivial extension B = C x E, we give necessary and sufficient conditions for each con to be surjective. We prove the surjectivity of phi(1) for a class of trivial extensions that includes relation extensions and hence cluster -tilted algebras. Finally, we study the kernel of phi(1) for any trivial extension, and give a more precise description of this kernel in the case of relation extensions. (c) 2015 Elsevier B.V. All rights reserved.