Koszul duality and the PBW theorem in symmetric tensor categories in positive characteristic

Abstract

We generalize the theory of Koszul complexes and Koszul algebras (in particular, Koszul duality between symmetric and exterior algebras) to symmetric tensor categories. In characteristic $p\ge 5$, this theory exhibits peculiar effects, not observed in the classical theory. In particular, we show that the symmetric and exterior algebras of a non-invertible simple object in the Verlinde category ${\rm Ver}_p$ are almost Koszul (although not Koszul), and show how this gives examples of $(r,s)$-Koszul algebras with any $r,s\ge 2$. We also develop a theory of Lie algebras in symmetric tensor categories. We show that the PBW theorem may fail in ${\rm Ver}_p$, but it holds if one assumes a certain identity of degree $p$ which we call the $p$-Jacobi identity. This identity is a generalization to $p\ge 5$ of the identity $[x,x]=0$ required for Lie algebras in characteristic $2$ and the identity $[[x,x],x]=0$ for odd $x$ required for Lie superalgebras in characteristic $3$.