Minimal extensions of Tannakian categories in positive characteristic

Abstract

We extend [DGNO1, Theorem 4.5] and [LKW, Theorem 4.22] to positive characteristic (i.e., to the finite, not necessarily fusion, case). Namely, we prove that if D is a finite nondegenerate braided tensor category over an algebraically closed field k of characteristic p > 0, containing a Tannakian Lagrangian subcategory Rep(G), where G is a finite k-group scheme, then D is braided tensor equivalent to Rep(D ω (G)) for some ω ∈ H 3 (G, G m ), where D ω (G) denotes the twisted double of G [G]. We then prove that the group M ext (Rep(G)) of minimal extensions of Rep(G) is isomorphic to the group H 3 (G, G m ). In particular, we use [EG2, FP] to show that M ext (Rep(µ p )) = 1, M ext (Rep(α p )) is infinite, and if O(Γ) * = u(g) for a semisimple restricted p-Lie algebra g, then M ext (Rep(Γ)) = 1 and M ext (Rep(Γ × α p )) ∼ = g * (1) .

Exact factorizations and extensions of finite tensor categories

“…Equivalently, Coh(G, ω) is the tensor category Rep (O(G), ω) of finite dimensional representations of the quasi-Hopf algebra (O(G), ω). (See [EG2,G3] for examples of nontrivial 3-cocycles on non constant finite group schemes. )…”