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

2016/03/26 by Etingof, Pavel · 3 citations
#Category Theory (math.CT) #FOS: Mathematics #Quantum Algebra (math.QA) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.1603.08133

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≥ 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≥ 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≥ 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.

Cited by

• Group schemes and their Lie algebras over a symmetric tensor category
• Homogeneous spaces in tensor categories
• Chevalley-Eilenberg cohomology of linearly reductive Lie algebras in the Verlinde category

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