# When is a group algebra Koszul?

Let $$KG$$ be a group algebra of a finite group $$G$$ such that the characteristic of $$K$$ divides the group order.

Question: When is a block of a group algebra (or even the whole group aglebra) a Koszul algebra? Is there a classification?

For example in characteristic two, the group algebra $$KG$$ is isomorphic to $$K[x]/(x^2)$$ when $$G$$ has two elements and here the group algebra is a Koszul algebra with Koszul dual the polynomial ring $$K[x]$$.

For a finite group $$G$$ and a field $$K$$ of characteristic $$p$$ dividing $$|G|$$, the group algebra $$KG$$ is Koszul if and only if $$p=2$$ and $$G/O(G)$$ is an elementary abelian $$2$$-group. This is because if $$KG$$ is Koszul then Ext over the cohomology ring is finite total dimension, so $$H^*(G,K)$$ is regular, and hence a polynomial ring. This happens if and only if $$G/O(G)$$ is an elementary abelian $$2$$-group, by a theorem I proved with Jon Carlson (MR1142778). Here, $$O(G)$$ denotes the largest odd order normal subgroup of $$G$$.
• Incidentally, when the group is not a $p$-group, I'm not quite sure how you want to interpret the word "Koszul", so maybe the answer is really if and only if $p=2$ and $G$ is an elementary abelian $2$-group. Mar 26 at 13:42
• What do you mean? I use the definition of Koszul algebra as in the article "KOSZUL DUALITY PATTERNS IN REPRESENTATION THEORY" by Beilinson, Ginzburg and Soergel, so that $A_0$ has just to be semisimple.
• They are using ${\mathbb Z}$-graded algebras with zero part semisimple. Are you applying their definition to the associated graded of the radical filtration? Mar 26 at 16:47