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

TL;DR

This paper extends Koszul duality and PBW theorem concepts to symmetric tensor categories in positive characteristic, revealing unique phenomena and conditions under which classical results hold or fail.

Contribution

It generalizes Koszul theory to symmetric tensor categories in characteristic p ≥ 5 and introduces a new p-Jacobi identity affecting the PBW theorem.

Findings

Symmetric and exterior algebras in Ver_p are almost Koszul.

Examples of (r,s)-Koszul algebras with r,s ≥ 2 are constructed.

The PBW theorem can fail in Ver_p unless the p-Jacobi identity holds.

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 $3$.