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

Abstract

We generalize the theory of Koszul complexes and Koszul algebras to symmetric tensor categories. In characteristic zero the generalization is routine, while in characteristic p there is a subtlety – the symmetric algebra of an object is not always Koszul (i.e., its Koszul complex is not always exact). Namely, this happens in the Verlinde category Verp in any characteristic p≥5. We call an object Koszul if its symmetric algebra is Koszul, and show that the only Koszul objects of Verp are usual supervector spaces, i.e., a non-invertible simple object Lm (2≤m≤p−2) is not Koszul. We show, however, that the symmetric algebra SLm is almost Koszul in the sense of Brenner, Butler and King (namely, (p−m,m)-Koszul), and compute the corresponding internal Yoneda algebra (i.e., the internal Ext-algebra from the trivial module to itself).We then proceed to discuss the PBW theorem for operadic Lie algebras (i.e., algebras over the operad Lie). This theorem is well known to fail for vector spaces in characteristic 2 (as one needs to require that [x,x]=0), and for supervector spaces in characteristic 3 (as one needs to require that [[x,x],x]=0 for odd x), but it holds in these categories in any characteristic p≥5; there is a well known proof based on Koszul duality. However, we show that in the category Verp, because of failure of Koszul duality, the PBW theorem can fail in any characteristic p≥5. Namely, one needs to impose the p-Jacobi identity, a certain generalization to characteristic p of the identities [x,x]=0 and [[x,x],x]=0. On the other hand, our main result is that once the p-Jacobi identity is imposed, the PBW theorem holds. This shows that the correct definition of a Lie algebra in Verp is an algebra over Lie which satisfies the p-Jacobi identity. This also applies to any symmetric tensor category that admits a symmetric tensor functor to Verp (e.g., a symmetric fusion category, see [19], Theorem 1.5). Finally, we prove the PBW theorem for Lie algebras in any quasi-semisimple symmetric tensor category.