Mini-Workshop: Cohomology of Hopf Algebras and Tensor Categories
Symmetry of the Mackey algebra
Mackey functors were introduced as a convienient tool for handling the induction theory of several objects having a similar behaviour (group representations; representation rings, group cohomology, etc;). Later, it was proved by Thévenaz and Webb that the category of Mackey functors is equivalent to the category of modules over a finite dimensional algebra called the Mackey algebra. The proof is far from being difficult, but this result is of crucial importance : one can study Mackey functors using ring and module theory. It turns out that the Mackey algebra is, in many aspects, similar to the group algebra but there are many interesting differences.
Let \(k\) is a field of characteristic \(p\) and \(G\) be a finite group. We denote by \(kG\) the group algebra of \(G\), by \(\mu_k(G)\) its Mackey algebra and by \(co\mu_k(G)\) the cohomological Mackey algebra of \(G\) which is an interesting quotient of the Mackey algebra.
| Property | \(kG\) | \(\mu_k(G)\) | \(co\mu_k(G)\) |
|---|---|---|---|
| Dimension | \(|G|\) | independant of \(k\) | \(\sum_{H,K\leq G}|H\backslash G/K|\) |
| Semisimple | \(p \not| |G|\) | \(p \not| |G|\) | \(p \not| |G|\) |
| Symmetric | Yes | \(p^2 \not| |G|\) | \(p \not| |G|\) |
| Finite Rep type | Sylow Cyclic | \(p^2 \not| |G|\) | \(p^2 \not| |G|\) |
| Gorenstein | Yes | \(p^2 \not| |G|\) | Sylow cyclic or Dihedral |
Here, an algebra is said to be Gorenstein if any finitely generated projective module has a finite injective resolution. It is well-known that the group algebras of finite representation are product of semisimple algberas and Brauer trees algebras. This is also true for the Mackey algebras but not for the cohomological Mackey algebras. The cohomological Mackey algebra is, in this case a gendosymmetric algebra of finite representation type.
Another interesting property of the categories of Mackey functors is that they have a `canonical' structure of closed symmetric monoidal category. By canonical, we mean that it extends the cartesian product of \(G\)-sets. Note that the Mackey algebras are not Hopf algebras in general. The monoidal structure can be constructed by Day convolution or by using an associative tri-module over the Mackey algebra.
In this talk, using this monoidal structure, I will explain how we can build central linear forms on the Mackey algebra. This leads to a characterization of the Mackey algebras that are symmetric.