On Monoidal Autoequivalences of the Category of Yetter-Drinfeld Modules Over a Group: The Lazy Case

Lentner, Simon; Priel, Jan

doi:10.1007/s10468-018-9809-1

Cited by 5 publications

(19 citation statements)

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“…In [LP15b] we have proved that every element fulfilling an additional condition (laziness) decomposes accordingly into an ordered product in these subgroups, also we have checked the Brauer-Picard group in known cases by hand. The Brauer-Picard group decomposition retains roughly the properties that a Lie group over a ring admits (not an honest Bruhat decomposition), which is what we get e.g.…”

Section: Motivationmentioning

confidence: 90%

“…The main motivation for our initial work [LP15b] was the case H = C[G] for G abelian, as treated in the second authors joint paper [FPSV14]. In particular let G ∼ = Z n p with p a prime number.…”

Section: Motivationmentioning

confidence: 99%

“…This additional condition is (see e.g. [LP15b]) equivalent to so-called laziness. In particular the extension G is isomorphic to G by the trivial isomorphism (identity on G /S and the nondegenerate form defined by α identifying S ∼ =Ŝ) and the additional morphism f is actually a Hopf algebra isomorphism induced by a group isomorphism v : G → G .…”

Section: Similarly One Constructs Vice-versamentioning

confidence: 99%

“…In [LP15b] we have proposed an approach to calculate BrPic(C) for C = H-mod by defining certain natural subgroups 1 BV, EV with intersection V and a set of elements R, such that the Brauer Picard group may decompose as a Bruhat-alike decomposition The automorphism group of an object C is the group of monoidal autoequivalences Eq mon (C) = Aut mon (C) resp. braided autoequivalences Eq br (Z) = Aut br (Z).…”

Section: Introductionmentioning

confidence: 99%

“…This "lazy" here is much less critical than in[LP15b], where we classify lazy braided autoequivalences of the Drinfeld center. In the present approach it is merely a technical inconvenience that we have good explicit formulae only for (still non-lazy) induction from a lazy monoidal autoequivalence of Rep(G).…”

mentioning

confidence: 96%

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

Lentner¹,

Priel²

2017

Bull. Belg. Math. Soc. Simon Stevin

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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...

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Section: Motivationmentioning

confidence: 90%

Section: Motivationmentioning

confidence: 99%

Section: Similarly One Constructs Vice-versamentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 96%

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

Lentner¹,

Priel²

2017

Bull. Belg. Math. Soc. Simon Stevin

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Homomorphisms and Rigid Isomorphisms of Twisted Group Doubles

Keilberg¹

2019

Algebr Represent Theor

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A decomposition of the Brauer–Picard group of the representation category of a finite group

Lentner

Priel

2017

Journal of Algebra

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Abstract. We present an approach of calculating the group of braided autoequivalences of the category of representations of the Drinfeld double of a finite dimensional Hopf algebra H and thus the Brauer-Picard group of H-mod. We consider two natural subgroups and a subset as candidates for generators. In this article H is the group algebra of a finite group G. As our main result we prove that any element of the Brauer-Picard group, fulfilling an additional cohomological condition, decomposes into an ordered product of our candidates. For elementary abelian groups G our decomposition reduces to the Bruhat decomposition of the Brauer-Picard group, which is in this case a Lie group over a finite field. Our results are motivated by and have applications to symmetries and defects in 3d-TQFT and group extensions of fusion categories.

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On Monoidal Autoequivalences of the Category of Yetter-Drinfeld Modules Over a Group: The Lazy Case

Cited by 5 publications

References 21 publications

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

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

Homomorphisms and Rigid Isomorphisms of Twisted Group Doubles

A decomposition of the Brauer–Picard group of the representation category of a finite group

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