On Frobenius exact symmetric tensor categories

2018

We develop the theory of semisimplifications of tensor categories defined by Barrett and Westbury. In particular, we compute the semisimplification of the category of representations of a finite group in characteristic p in terms of representations of the normnalizer of its Sylow p-subgroup. This allows us to compute the semisimplification of the representation category of the symmetric group S_n+p in characteristic p, where 0< n< p-1, and of the Deligne category Rep^ abS_t, where t∈ N. We also compute the semisimplification of the category of representations of the Kac-De Concini quantum group of the Borel subalgebra of sl_2. We also study tensor functors between Verlinde categories of semisimple algebraic groups arising from the semisimplification construction, and objects of finite type in categories of modular representations of finite groups (i.e., objects generating a fusion category in the semisimplification). Finally, we determine the semisimplifications of the tilting...

arXiv (Cornell University), 2020

We propose a method of constructing abelian envelopes of symmetric rigid monoidal Karoubian categories over an algebraically closed field k. If char(k) = p > 0, we use this method to construct generalizations Verpn, Ver + p n of the incompressible abelian symmetric tensor categories defined in [5] for p = 2 and in [28, 30] for n = 1. Namely, Verpn is the abelian envelope of the quotient of the category of tilting modules for SL2(k) by the n-th Steinberg module, and Ver + p n is its subcategory generated by P GL2(k)-modules. We show that Verpn are reductions to characteristic p of Verlinde braided tensor categories in characteristic zero, which explains the notation. We study the structure of these categories in detail, and in particular show that they categorify the real cyclotomic rings Z[2 cos(2π/p n)], and that Verpn embeds into Ver p n+1. We conjecture that every symmetric tensor category of moderate growth over k admits a fiber functor to the union Verp∞ of the nested sequence Verp ⊂ Ver p 2 ⊂ • • •. This would provide an analog of Deligne's theorem in characteristic zero and a generalization of the results of [16] which shows that this conjecture holds for Frobenius exact (in particular, semisimple) categories, and moreover the fiber functor lands in Verp (in the case of fusion categories, this was shown earlier in [35]). Finally, we classify symmetric tensor categories generated by an object with invertible exterior square; this class contains the categories Verpn. 4.1. The definition of the Verlinde categories Ver p n , Ver + p n 32 4.2. T n,p is complete 32 4.3. Basic properties of Ver p n 34 4.4. The lift of Ver p n to characteristic zero 35 4.5. The Frobenius-Perron dimension and the Grothendieck ring of Ver p n 36 4.6. The Cartan matrix and the decomposition matrix of Ver p n 36 4.7. An explicit formula for the Cartan matrix of Ver p n 38 4.8. The objects T i = F (T i) 40 4.9. Some properties of the Frobenius functor 41 4.10. The inclusion Ver p n−1 ֒→ Ver p n 42 4.11. The tensor product theorem for Ver p n 43 4.12. Tensor product of simple objects 45 4.13. The blocks of Ver p n 46 4.14. The principal block 47 4.15. Ver p n is a Serre subcategory in Ver p n+k 48 4.16. Ext 1 between simple objects 48 4.17. The Grothendieck ring of the stable category 50 4.18. Incompressibility 51 4.19. Classification of symmetric tensor categories generated by an object with invertible exterior square 53 5. Examples 54 5.1. The category Ver p 2 54 5.2. The category Ver + p 3 in the case p = 3 57 References 58

Mathematical Research Letters, 2011

Let k be any field. J-P. Serre proved that the spectrum of the Grothendieck ring of the k−representation category of a group is connected, and that the same holds in characteristic zero for the representation category of a Lie algebra over k [Se]. We say that a tensor category C over k is virtually indecomposable if its Grothendieck ring contains no nontrivial central idempotents. We prove that the following tensor categories are virtually indecomposable: Tensor categories with the Chevalley property; representation categories of affine group schemes; representation categories of formal groups; representation categories of affine supergroup schemes (in characteristic = 2); representation categories of formal supergroups (in characteristic = 2); symmetric tensor categories of exponential growth in characteristic zero. In particular, we obtain an alternative proof to Serre's Theorem, deduce that the representation category of any Lie algebra over k is virtually indecomposable also in positive characteristic (this answers a question of Serre [Se]), and (using a theorem of Deligne [D] in the super case, and a theorem of Deligne-Milne [DM] in the even case) deduce that any (super)Tannakian category is virtually indecomposable (this answers another question of Serre [Se]).

Journal of Algebra, 2005

We consider the problem of decomposing tensor powers of the fundamental level 1 highest weight representation V of the affine Kac-Moody algebra g(E 9 ). We describe an elementary algorithm for determining the decomposition of the submodule of V ⊗n whose irreducible direct summands have highest weights which are maximal with respect to the null-root. This decomposition is based on Littelmann's path algorithm and conforms with the uniform combinatorial behavior recently discovered by H. Wenzl for the series EN , N = 9. 2

2002

Sei V ein endlich-dimensionaler, komplexer Vektorraum. Eine Teilmenge X in V hat die Trennungseigenschaft, falls das Folgende gilt: Für je zwei linear unabhängige lineare Funktionen l, m auf V existiert ein Punkt x in X mit l(x) = 0 und m(x) = 0. Wir interessieren uns für den Fall V = C[x, y] n , d.h. V ist eine irreduzible Darstellung von SL 2 . Die Teilmengen, die wir untersuchen, sind Bahnabschlüsse von Elementen aus C[x, y] n . Wir beschreiben die Bahnen, die die Trennungseigenschaft erfüllen:

Moscow Mathematical Journal

We start the general structure theory of not necessarily semisimple finite tensor categories, generalizing the results in the semisimple case (i.e. for fusion categories), obtained recently in our joint work with D. Nikshych. In particular, we generalize to the categorical setting the Hopf and quasi-Hopf algebra freeness theorems due to Nichols-Zoeller and Schauenburg, respectively. We also give categorical versions of the theory of distinguished group-like elements in a finite dimensional Hopf algebra, of Lorenz's result on degeneracy of the Cartan matrix, and of the absence of primitive elements in a finite dimensional Hopf algebra in zero characteristic. We also develop the theory of module categories and dual categories for not necessarily semisimple finite tensor categories; the crucial new notion here is that of an exact module category. Finally, we classify indecomposable exact module categories over the simplest finite tensor categories, such as representations of a finite group in positive characteristic, representations of a finite supergroup, and representations of the Taft Hopf algebra.

arXiv (Cornell University), 2023

Journal of K-theory: K-theory and its Applications to Algebra, Geometry, and Topology, 2014

In this paper we begin studying tannakian constructions in ∞-categories and combine them with the theory of motivic categories developed by Hanamura, Levine, and Voevodsky. This paper is the first in a series of papers. For the purposes above, we first construct a derived affine group scheme and its representation category from a symmetric monoidal ∞-category, which we shall call the tannakization of a symmetric monoidal ∞-category. It can be viewed as an ∞-categorical generalization of work of Joyal-Street and Nori. Next we apply it to the stable ∞-category of mixed motives equipped with the realization functor of a mixed Weil cohomology. We construct a derived motivic Galois group which represents the automorphism group of the realization functor, and whose representation category satisfies an appropriate universal property. As a consequence, we construct an underived motivic Galois group of mixed motives, which is a pro-algebraic group and has nice properties. Also, we present ba...

Quantum Topology, 2018

To every object X of a symmetric tensor category over a field of characteristic p > 0 we attach p-adic integers Dim + (X) and Dim -(X) whose reduction modulo p is the categorical dimension dim(X) of X, coinciding with the usual dimension when X is a vector space. We study properties of Dim ± (X), and in particular show that they don't always coincide with each other, and can take any value in Z p . We also discuss the connection of p-adic dimensions with the theory of λ-rings and Brauer characters.

2006

For a rigid tensor abelian category $T$ over a field $k$ we introduce a notion of a normal quotient $q:T\to Q$. In case $T$ is a Tannaka category, our notion is equivalent to Milne's notion of a normal quotient. More precisely, if $T$ is the category of finite dimensional representations of a groupoid scheme $G$ over $k$, then $Q$ is equivalent to the representation category of a normal subgroupoid scheme of $G$. We describe such a quotient in terms of the subcategory $S$ of $T$ consisting of objects which become trivial in $Q$. We show that, under some condition on $S$, $Q$ is uniquely determined by $S$. If $S$ is an 'etale finite tensor category, we show that the quotient of $T$ by $S$ exists. In particular we show the existence of the base change of $T$ with respect to finite separable field extensions. As an application, we obtain a condition for the exactness of sequences of groupoid schemes in terms of the representation categories.