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 φ⁎:HH⁎(B)→HH⁎(C)φ⁎: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 Schiffler.In the case of a trivial extension B=C⋉EB=C⋉E, we give necessary and sufficient conditions for each φnφn to be surjective. We prove the surjectivity of φ1φ1 for a class of trivial extensions that includes relation extensions and hence cluster-tilted algebras. Finally, we study the kernel of φ1φ1 for any trivial extension, and give a more precise description of this kernel in the case of relation extensions.