Descent and Forms of Tensor Categories

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

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...

Communications in Algebra, 2011

We study good (i.e., semisimple) reductions of semisimple rigid tensor categories modulo primes. A prime p is called good for a semisimple rigid tensor category C if such a reduction exists (otherwise, it is called bad). It is clear that a good prime must be relatively prime to the Müger squared norm |V | 2 of any simple object V of C. We show, using the Ito-Michler theorem in finite group theory, that for group-theoretical fusion categories, the converse is true. While the converse is false for general fusion categories, we obtain results about good and bad primes for many known fusion categories (e.g., for Verlinde categories). We also state some questions and conjectures regarding good and bad primes.

Journal of Pure and Applied Algebra, 2014

In this paer, we compute the Clebsch-Gordan formulae and the Green rings of connected pointed tensor categories of finite type.

Applied Categorical Structures, 2008

We establish an algebra-isomorphism between the complexified Grothendieck ring F of certain bimodule categories over a modular tensor category and the endomorphism algebra of appropriate morphism spaces of those bimodule categories. This provides a purely categorical proof of a conjecture by Ostrik concerning the structure of F . As a by-product we obtain a concrete expression for the structure constants of the Grothendieck ring of the bimodule category in terms of endomorphisms of the tensor unit of the underlying modular tensor category.

2014

We survey some results on tensor products of irreducible Harish-Chandra bimodules. It turns out that such tensor products are semisimple in suitable Serre quotient categories. We explain how to identify the resulting semisimple tensor categories and describe some applications to representation theory.

Mathematical Research Letters, 2009

We prove a general result which implies that the global and Frobenius-Perron dimensions of a fusion category generate Galois invariant ideals in the ring of algebraic integers.

arXiv: Category Theory, 2016

We define a tensor product of linear sites, and a resulting tensor product of Grothendieck categories based upon their representations as categories of linear sheaves. We show that our tensor product is a special case of the tensor product of locally presentable linear categories, and that the tensor product of locally coherent Grothendieck categories is locally coherent if and only if the Deligne tensor product of their abelian categories of finitely presented objects exists. We describe the tensor product of non-commutative projective schemes in terms of Z-algebras, and show that for projective schemes our tensor product corresponds to the usual product scheme.

Journal of Noncommutative Geometry, 2012

This paper addresses the problem of describing the structure of tensor C*-categories M with conjugates and irreducible tensor unit. No assumption on the existence of a braided symmetry or on amenability is made. Our assumptions are motivated by the remark that these categories often contain non-full tensor C*-subcategories with conjugates and the same objects admitting an embedding into the Hilbert spaces. Such an embedding defines a compact quantum group by Woronowicz duality. An important example is the Temperley-Lieb category canonically contained in a tensor C*-category generated by a single real or pseudoreal object of dimension 2. The associated quantum groups are the universal orthogonal quantum groups of Wang and Van Daele. Our main result asserts that there is a full and faithful tensor functor from M to a category of Hilbert bimodule representations of the compact quantum group. In the classical case, these bimodule representations reduce to the G-equivariant Hermitian bundles over compact homogeneous G-spaces, with G a compact group. Our structural results shed light on the problem of whether there is an embedding functor of M into the Hilbert spaces. We show that this is related to the problem of whether a classical compact Lie group can act ergodically on a non-type I von Neumann algebra. In particular, combining this with a result of Wassermann shows that an embedding exists if M is generated by a pseudoreal object of dimension 2.

Journal of Mathematical Physics, 2015

In this paper, we describe a Mackey type decomposition for group actions on abelian categories. This allows us to define new Mackey functors which associates to any subgroup the K-theory of the corresponding equivariantized abelian category. In the case of an action by tensor autoequivalences, the Mackey functor at the level of Grothendieck rings has a Green functor structure. As an application we give a description of the Grothendieck rings of equivariantized fusion categories under group actions by tensor autoequivalences on graded fusion categories. In this settings, a new formula for the tensor product of any two simple objects of an equivariantized fusion category is given, simplifying the fusion formula from Burciu and Natale [J. Math. Phys. 54, 013511 (2013)].