I am seeking references to places where operads and their algebras have been studied for the stable module category. Colored operads are fine too.
Let $k$ be a field and $R$ a $k$-algebra. The stable module category StMod(R) is the category of R-modules where we've modded out by all morphisms that factor through a projective. This triangulated category is useful in representation theory (see Happel's book) and for the study of Tate Cohomology (see work of Benson, Carlson, Rickard, Iyengar, Christensen, Minac, Beligiannis, Gillespie, Bravo, Stovicek, Krause, Pevtsova). The stable module category is also the homotopy category of a stable model structure on $R$-mod (shown originally by Pirashvili then by Hovey in his book). This is a monoidal model category when $R = k[G]$ for a finite group $G$ (see section 9 of Hovey's "Cotorsion pairs, model category structures, and representation theory), and this is the setting I am interested in.
Right now the only examples of operad-algebras in this category I can think of are:
- Algebras over $Ass$ are associative $R$-algebras
- Algebras over $Com$ are commutative $R$-algebras
- Algebras over $Lie$ should work here and be $R$-algebras that additionally have a Lie structure.
My reason is that, motivated by applications in Top and Ch(k), I recently did some model categorical work on the interplay between localization and colored operads. I've realized the model structure on $R$-mod from above also satisfies all my hypotheses, so there might be applications of my work in this setting. Obviously, I need to read up first on what has been done and what sorts of questions people care about, hence the reference request. The first two examples above seem intrinsically interesting, but I don't know if their homotopy theory has ever been studied elsewhere. I know almost nothing about the third example. References where these three examples or others have been studied in StMod(R) would help me figure out what is currently known and what sorts of questions are the right questions to work on in this area.