On symmetric fusion categories in positive characteristic
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On symmetric fusion categories in positive characteristic Victor Ostrik1,2
© Springer Nature Switzerland AG 2020
Abstract We propose a conjectural extension in the positive characteristic case of well known Deligne’s theorem on the existence of super fiber functors. We prove our conjecture in the special case of semisimple categories with finitely many isomorphism classes of simple objects. Mathematics Subject Classification 18M05 · 18M20
1 Introduction 1.1 Pre-Tannakian categories Let k be an algebraically closed field of characteristic p ≥ 0. We recall that a symmetric monoidal category C is a category endowed with the functor ⊗ : C × C → C of tensor product, with associativity and commutativity isomorphisms and unit object 1 satisfying suitable axioms, see e.g. [25] or [15, Definition 8.1.12]. Such a category is called a symmetric tensor category if the following holds: 1) C is a k−linear abelian category; 2) the functor ⊗ is k−bilinear and the natural morphism k → EndC (1) is an isomorphism; 3) C is rigid (this implies that the functor ⊗ is exact in each variable, see [7, Proposition 1.16] or [15, Proposition 4.2.1]). In this paper we are interested in symmetric tensor categories C satisfying the following assumptions:
To the memory of my parents, Rabina Tatiana Fedorovna and Rabin Vladimir Akimovich.
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Victor Ostrik [email protected]
1
Department of Mathematics, University of Oregon, Eugene, OR 97403, USA
2
Laboratory of Algebraic Geometry, National Research University Higher School of Economics, Moscow, Russia 0123456789().: V,-vol
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4) C is essentially small, any morphism space is finite dimensional and each object has finite length. Such categories are precisely tensor categories satisfying finiteness assumptions of [8, 2.12.1]; they were called pre-Tannakian in [6, 2.1]. Example 1.1 (i) The category Vec of finite dimensional vector spaces is preTannakian. Now let p = 2. Then the category sVec of finite dimensional super vector spaces over k is pre-Tannakian. (ii) Let G be an affine group scheme over k. Then the category Repk (G) of finite dimensional representations of G over k is pre-Tannakian. (iii) (see [9, 0.3]) Let G be an affine super group scheme over k and let ε ∈ G(k) be an element of order ≤ 2 such that its action by conjugation on G coincides with the parity automorphism of G. Let Repk (G, ε) be the full subcategory of super representations of G such that ε acts by the parity automorphism. Then Repk (G, ε) is pre-Tannakian. A special case of this construction is when G is a finite group and ε ∈ G is a central element of order ≤ 2, see [9, 0.4 (i)]. (iv) (see [8, Section 8]) Let C be a pre-Tannakian category and let π ∈ C be its fundamental group as defined in [8, 8.13]. Thus π is an affine group scheme in the category C and it acts on any object of C in a canonical way. Let G be an affine group scheme in the category C and let ε : π → G be a homomorphism such that the action of π on G by conjugations coincides with th
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