Autoequivalences of Tensor Categories Attached to Quantum Groups at Roots of 1

Abstract

We compute the group of braided tensor autoequivalences and the Brauer-Picard group of the representation category of the small quantum group u q (g), where q is a root of unity.To the memory of Bertram Kostant IntroductionLet k be an algebraically closed field of characteristic zero. Let G be a simple algebraic group over k, and let g = Lie(G) be its Lie algebra. Let q be a root of unity of odd order coprime to 3 if G is of type G 2 , and coprime to the determinant of the Cartan matrix of G. Let u q (g) be Lusztig's small quantum group attached to g [Lu1]. Then u q (g) is a quasitriangular Hopf algebra, so the category of its finite dimensional representationsRep u q (g) is a finite braided tensor category [EGNO]. One of the main goals of this paper is to compute the Picard group Pic(Rep u q (g)) of this category, i.e., the group of equivalence classes of invertible Rep u q (g)-module categories. Picard groups of braided tensor categories and, in particular, Brauer-Picard groups of tensor categories play a crucial role in classification of graded extensions [ENO] and also appear as symmetry groups of three-dimensional topological field theories [FS]. It is known that Pic(Rep u q (g)) is isomorphic to the group Aut br (Rep u q (g)) of braided autoequivalences of Rep u q (g) [DN, ENO]. We show under some restrictions on q that Aut br (Rep u q (g)) is isomorphic to the group Aut(g) of automorphisms of g, i.e., Aut br (Rep u q (g)) = Γ ⋉ G ad , where G ad is the adjoint group of G and Γ = Γ g is the automorphism group of the Dynkin diagram of g. Namely, we prove this when the order ℓ of q is sufficiently large and also for classicalMoreover, we show that Rep u q (g) has only two braidings (the standard one and its reverse) and deduce that any tensor autoequivalence of Rep u q (g) is necessarily braided.Thus, the group of tensor autoequivalences (also known as the group of biGalois objects) of Rep u q (g) is isomorphic to Γ ⋉ G ad . This generalizes the result of Bichon [Bi2], who proved this fact for g = sl 2 .Date: March 21, 2017. 1 2 ALEXEI DAVYDOV, PAVEL ETINGOF, AND DMITRI NIKSHYCHWe also consider the braided tensor category O q (G) − comod of finite dimensional comodules over the function algebra O q (G), which is the G-equivariantization of Rep u q (g).We show that every braided autoequivalence of O q (G) − comod comes from a Dynkin diagram automorphism if ℓ is sufficiently large, and prove a similar result in the non-braided case. This generalizes a result of Neshveyev and Tuset [NT1, NT2], who proved this when q is not a root of unity. We also show this for the classical groups SL N , Sp N , SO N if ℓ > N.As a tool, we introduce the notion of a finitely dominated tensor category. We show that the category of comodules over a finitely presented Hopf algebra is finitely dominated and prove that tensor autoequivalences of a finitely dominated category that preserve a tensor generator form an algebraic group. While this theory plays an auxiliary role in our paper, it may be of independent interest. We expect t...

The Brauer–Picard groups of fusion categories coming from the ADE subfactors

“…At an initial glance it also appears that our results overlap with [5]. However the tensor categories we work with are different to the tensor categories the cited paper deals with.…”

“…We have Aut br (Z(Ad(E 6 ))) = Z 2Z. By considering dimensions and twists, we can see that there is only one possible non-trivial fusion ring automorphism of Z(Ad(E 6 )), that exchanges the objects 1 2 (f (5) ⊠ f (1) + S) and…”

“…As the Ad(A N ) planar algebra has the single generator T ∈ P 3 it is enough to consider how φ behaves on T . As P 3 is one dimensional it follows that φ(T ) = αT for some α ∈ C. As φ must fix the single strand we can apply φ to relation (5) to show that α ∈ {1, −1}. It can be verified that φ(T ) = −T is consistent with the other relations and hence determines a valid planar algebra automorphism.…”

Lie theory for fusion categories: A research primer

“…(Simple factorizations of C(sl p , k)) For primes p, the factorization in (16) into simple components is easily described for C(sl p , k) when p ∤ k. In these cases the pointed subcategory has nontrivial simple objects kλ i for 1 ≤ i ≤ p − 1 whose fusion rules have the structure of the cyclic group Z/pZ. The quadratic form [22,Section 8.4] of the pointed subcategory is determined by the twists θ kλi (see (17) below) which imply the form is degenerate if and only if p | k. As p is prime, C(sl p , k) pt is simple, and by Sawin's classification of fusion subcategories C(sl p , k) ′ pt is simple as well. When p | k, θ kλi = 1 for all 1 ≤ i ≤ n − 1 and C(g, k) pt ≃ Rep(Z/pZ) which is symmetrically braided.…”

“…Example 18 (Modular data of C(sl 2 , ℓ, q)). The Weyl group W is isomorphic to Z/2Z so by (17) we have θ sλ = q s(s+2)/2 and…”