I haven't seen this question in standard textbooks, so I decide to give it a try here. It might relate to deeper structures of certain TQFTs, but I'm not sure.

Let $G$ be a finite group. Its finite-dimensional complex representations are well understood, the regular representation $\mathbb{C}[G]$ being the most important one. Note that it's not only a vector space acted by $G$: it also has a natural algebra structure and naturally compatible with $G$-action (EDIT: .. in a less natural fashion --- $g(ab)= (ga)b$, instead of $g(ab) = gagb$.)

###Question Is there a classification of group representations that have compatible algebra structures?

I haven't seen this question in standard textbooks, so I decide to give it a try here. It might relate to deeper structures of certain TQFTs, but I'm not sure.

Let $G$ be a finite group. Its finite-dimensional complex representations are well understood, the regular representation $\mathbb{C}[G]$ being the most important one. Note that it's not only a vector space acted by $G$: it also has a natural algebra structure and naturally compatible with $G$-action (EDIT: .. in a less natural fashion --- $g(ab)= (ga)b$, instead of $g(ab) = gagb$.)

Question

Is there a classification of group representations that have compatible algebra structures?

I haven't seen this question in standard textbooks, so I decide to give it a try here. It might relate to deeper structures of certain TQFTs, but I'm not sure.

Let $G$ be a finite group. Its finite-dimensional complex representations are well understood, the regular representation $\mathbb{C}[G]$ being the most important one. Note that it's not only a vector space acted by $G$: it also has a natural algebra structure and naturally compatible with $G$-action.

###Question Is there a classification of group representations that have compatible algebra structures?

I haven't seen this question in standard textbooks, so I decide to give it a try here. It might relate to deeper structures of certain TQFTs, but I'm not sure.

Let $G$ be a finite group. Its finite-dimensional complex representations are well understood, the regular representation $\mathbb{C}[G]$ being the most important one. Note that it's not only a vector space acted by $G$: it also has a natural algebra structure and naturally compatible with $G$-action (EDIT: .. in a less natural fashion --- $g(ab)= (ga)b$, instead of $g(ab) = gagb$.)

###Question Is there a classification of group representations that have compatible algebra structures?

I haven't seen this question in standard textbooks, so I decide to give it a try here. It might relate to deeper structures of certain TQFTs, but I'm not sure.

Let $G$ be a finite group. Its finite dimensional complex representions are well understood, the regular representation $\mathbb{C}[G]$ being the most important one. Note that its not only a vector space acted by $G$: it also has a natural algebra structure and naturally compatible with $G$ action.

###Question Is there a classification of group representations that have compatible algebra structures?

I haven't seen this question in standard textbooks, so I decide to give it a try here. It might relate to deeper structures of certain TQFTs, but I'm not sure.

Let $G$ be a finite group. Its finite-dimensional complex representations are well understood, the regular representation $\mathbb{C}[G]$ being the most important one. Note that it's not only a vector space acted by $G$: it also has a natural algebra structure and naturally compatible with $G$-action.

###Question Is there a classification of group representations that have compatible algebra structures?