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) 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⋉E, we give necessary and sufficient conditions for each φn to be surjective. We prove the surjectivity of φ1 for a class of trivial extensions that includes relation extensions and hence cluster-tilted algebras. Finally, we study the kernel of φ1 for any trivial extension, and give a more precise description of this kernel in the case of relation extensions.