TITLE: Are there Steenrod operations on Hochschild cohomology of the  group algebra of a finite group?
QUESTION [7 upvotes]: I know that the steenrod algebra acts on the group cohomology. I would like to know whether this action extends to the Hochschild cohomology of the group algebra, as the latter contains group cohomology as a subalgebra.

REPLY [2 votes]: Yes, there is an action of the Steenrod algebra on HH^*(k[G]; k[G]), G a discrete group and k = Z/2 for the mod 2 Steenrod algebra.  In my paper "A Comparison of Products in Hochschild cohomology," (on arXiv.org), I show how Steenrod's cup-i products act on the Hochschild cochain complex Hom_k(k[G]^*, k[G]) for an arbitrary coefficient ring k.  The cup-i products can then be used to define an action of the Sq^i operators.