I am trying to find more generalized counterparts of some well-known results from modular group representations.


My question is the following:


Suppose that $H$ is a finite $p$-group acting as automorphisms on a finite dimensional $k$-algebra $A$. Suppose moreover that the field $k$ has characteristic exactly this prime $p$. Does it follow that all simple $A\#kH$ modules are in bijection with all simple $A$-modules? Obviously, I am talking about isomorphism classes of simple modules.


In the case $A=kL$ for a $p'$-group $L$ I think this result is well known although I cannot identify a reference. At least a treatment of this case can be found in Section 8.4 of Webb's book:


<http://www-users.math.umn.edu/~webb/RepBook/RepBookLatex.pdf>


In this case, there is a canonical projection $A\# kH\rightarrow kH$ which makes $kH$ a canonical $A\# kH$-module. It turns out that this is a projective module and it is the projective cover of the trivial module.


Any reference to this type of smash products is very welcomed.



Representations of smash products with $p$-groups
Take $p=2$, $k$ algebraically closed, $A=kC\_3$, and $H=C\_2$ acting non-trivially on $C\_3$.


Then $A$ has three simple modules, but $A\# kH$ is the group algebra $kS\_3$, which has two simple modules.


