Three natural subgroups of the Brauer-Picard group of a Hopf algebra with applications

Abstract

In this article we construct three explicit natural subgroups of the Brauer-Picard group of the category of representations of a finite-dimensional Hopf algebra. In examples the Brauer Picard group decomposes into an ordered product of these subgroups, somewhat similar to a Bruhat decomposition.Our construction returns for any Hopf algebra three types of braided autoequivalences and correspondingly three families of invertible bimodule categories. This gives examples of so-called (2-)Morita equivalences and defects in topological field theories. We have a closer look at the case of quantum groups and Nichols algebras and give interesting applications. Finally, we briefly discuss the three families of group-theoretic extensions. 1 We choose these names EV, BV for compatibilities with previous conventions. Be advised that V does not necessarily have complement subgroups B, E in BV, EV in the most general cases. arXiv:1702.05133v1 [math.QA] 16 Feb 2017 2Example (Sec. 4.1.5). Let G ∼ = Z n p with p a prime number. Our decomposition reduces to the Bruhat decomposition of BrPic(Vect G ), which is the Lie group O 2n (F p ) over the finite field F p . In this case BV, EV are lower and upper triangular matrices, intersecting in the subgroup V = GL n (F p ). The partial dualizations are Weyl group elements. More precisely, our result reduces to the Bruhat decomposition of the Lie groups D n relative to the parabolic subsystem A n−1 , so reflections are actually equivalence classes corresponding to n + 1 cosets of the parabolic Weyl group.The present article is devoted to start the discussion of the more general case C = H-mod. We shall not try to prove a decomposition theorem, but focus our attention on establishing and discussing the expected natural subgroups V, BV, EV, R of the Brauer Picard group. We will also briefly discuss several interesting applications of our results, in particular when H is the Borel part of a quantum group resp. a Nichols algebra.In Section 2 we briefly recall the induction functor and the ENOM-functor [ENOM09]In view of interesting examples and the applications to defects in mathematical physics and Nichols algebras we state the obvious generalization of these concepts to the groupoid setting, so that arbitrary monoidal equivalences C ∼ → D give rise to invertible C-Dbimodule categories, and these are in bijection to braided equivalences Z(C)In Section 3 we define and derive for each subgroup V, BV, EV and the subset R explicit expressions for the braided autoequivalence as well as the invertible bimodule categories.On one hand BV resp. EV are obtained using induction functors from H-mod resp. H * -mod. So the bimodule categories in BrPic(H-mod) resp. BrPic(H * -mod) are given by definition. We then calculate explicitly the images under the ENOM functor using Bigalois objects and finally we describe again the preimage of EV now in BrPic(H-mod). As linear categories, the bimodule categories in BV are all equal to C, while the bimodule categories in EV are representation categories of Bigalois o...

’t Hooft Anomalies of Discrete Gauge Theories and Non-abelian Group Cohomology

“…The corresponding Brauer-Picard group for ω = 0 is studied in detail in [LP17a]. The more general case of the representation category of Hopf algebras which includes the case of non-trivial ω is studied in [LP17b]. The kinematical symmetries studied in this paper correspond to the subgroup of classical symmetries in [LP17b].…”

“…The more general case of the representation category of Hopf algebras which includes the case of non-trivial ω is studied in [LP17b]. The kinematical symmetries studied in this paper correspond to the subgroup of classical symmetries in [LP17b]. Three-dimensional G-equivariant extended field theories correspond to G-modular categories [TV14,SW18a].…”

A Family of New Borel Subalgebras of Quantum Groups

“…What additional structure is the implication of this fact? One might expect some weak link between the category O B and usual category O, maybe an invertible bimodule category as in [LP17]? (4) The induced modules appear for sl 2 in [Schm96,Tesch01] in the context of non-compact quantum group.…”