Are there Steenrod operations on Hochschild cohomology of the  group algebra of a finite group? 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.
 A: Quite unlikely. The many solutions to Deligne's conjecture for Hochschild cohomology show that the Hochschild complex is an $E_2$-algebra, and this is the best one can get. It is known that the cohomology of an $E_\infty$-algebra carries an action of the Steenrod algebra. If you wish, this is why the standard group cohomology is an algebra over the Steenrod algebra, since it is the cohomology of the (singular) cochains on a space, which is an $E_\infty$-algebra. The homology of an $E_2$-algebra has a very low-dimensional $p$-power operation, where $p$ is the cgaracteristic of the ground field, and that's essentially all, see the references below.
PD I've edited this comment to correct some inaccuracies.
A: In "An alegbraic approach to the Steenrod algebra" Peter May writes down pretty much all the situations where you could have the action of Steenrod "like" operations. The most relevant case here is that of the cohomology of a cocommutative Hopf algebra. In case you are unfamiliar with Hopf algebras, this mean that there is a product map $\mu :A \otimes A \to A$ and a coproduct map $\psi :A \to A \otimes A$. To be cocommutative $\psi=\tau \circ \psi$ where $\tau$ is the twist map. So it seems unlikely that there would be Steenrod type operations acting on the Hochschild cohomology of any algebra, but there are probably operations on the Hochschild cohomology of a cocommutative Hopf algebra. However, you would need to verify that if $A$ is a cocommutative Hopf algebra then $A \otimes A^{op}$ is as well. So that would insure that there are operations in $HH(A,k)$ but not in $HH(A)=HH(A,A)$. That I am not sure about.
A: 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.
