Invertible elements in $k[G]\otimes k[G]$

Question

Let $G$ be a finite group, $k$ a field and $k[G]$ the group algebra.

Is something known about the invertible elements in $k[G]\otimes k[G]$? Maybe the isomorphism $k[G]\otimes k[G]\cong k[G\times G]$ is usefull.

Answer 1

In general, classification of units in a group ring $k[H]$ is nontrivial (and the special case $H=G\times G$ isn't any easier.) You can find lots of material with diverse results online. Perhaps take a look at this writeup. As written, I don't think your question has a nice answer.

You have the trivial units, of course, but beyond that you'll just have to apply the partial results that are known to get restrictions on $H$, and then determine what that means for $G$ in $H=G\times G$.