Operations on the Secondary Hochschild Cohomology

We show that the secondary Hochschild cohomology associated to a triple $(A,B,\epsilon )$ has several of the properties of the usual Hochschild cohomology. Among others, we prove the existence of the cup and Lie products, discuss the connection with extensions of $B$-algebras, and give a Hodge type decomposition of the secondary Hochschild cohomology.