Constructing non-semisimple modular categories with local modules

Laugwitz, Robert; Walton, Chelsea

doi:10.48550/arxiv.2202.08644

Cited by 3 publications

(6 citation statements)

References 30 publications

(53 reference statements)

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“…For C rigid, the dual Verma module V * 0 in N N YD(C) is an algebra. This is the adjoint algebra in the sense of [Sh19] in the relative version [LW2,Mo22]. For highest-weight theory in the context of diagonal Nichols algberas see [Vay19].…”

Section: Splitting Statementsmentioning

confidence: 99%

A Kazhdan–Lusztig Correspondence for $$L_{-\frac{3}{2}}(\mathfrak {sl}_3)$$

Creutzig

Ridout

Rupert

2023

Commun. Math. Phys.

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Let U be a braided tensor category, typically unknown, complicated and in particular non-semisimple. We characterize U under the assumption that there exists a commutative algebra A in U with certain properties: Let C be the category of local A-modules in U and B the category of A-modules in U, which are in our set-up usually much simpler categories than U. Then we can characterize U as a relative Drinfeld center Z C (B) and B as representations of a certain Hopf algebra inside C.In particular this allows us to reduce braided tensor equivalences to the knowledge of abelian equivalences, e.g. if we already know that U is abelian equivalent to the category of modules of some quantum group U q or some generalization thereof, and if C is braided equivalent to a category of graded vector spaces, and if A has a certain form, then we already obtain a braided tensor equivalence between U and Rep(U q ).A main application of our theory is to prove logarithmic Kazhdan-Lusztig correspondences, that is, equivalences of braided tensor categories of representations of vertex algebras and of quantum groups. Here, the algebra A and the corresponding category C are a-priori given by a free-field realization of the vertex algebra and by a Nichols algebra. We illustrate this in those examples where the representation theory of the vertex algebra is well enough understood. In particular we prove the conjectured correspondences between the singlet vertex algebra M(p) and the unrolled small quantum groups of sl 2 at 2p-th root of unity. Another new example is the Kazhdan-Lusztig correspondence between the affine vertex algebra of gl 1|1 and an unrolled quantum group of gl 1|1 . * this should be q = exp πi r ∨ (k+h ∨ ) with h ∨ the dual Coxeter number of g and r ∨ the lacing number of the even subalgebra.

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Section: Splitting Statementsmentioning

confidence: 99%

A Kazhdan–Lusztig Correspondence for $$L_{-\frac{3}{2}}(\mathfrak {sl}_3)$$

Creutzig

Ridout

Rupert

2023

Commun. Math. Phys.

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“…ZpVect ω G q pAq over a rigid Frobenius algebra A as in Theorem 2 is a modular category by [LW22a,Theorem 4.12] and [KO02, Theorem 4.5] in the semisimple case. In fact, [DS17,Theorem 3.16] shows that such modular categories are equivalent as ribbon categories to ZpVect ω H{N q, for a 3-cocycle ω on H{N such that its pullback to H via the quotient homomorphism is equivalent to ω| H .…”

Section: The Category Of Local Modules Rep Locmentioning

confidence: 99%

“…Now assume |G : H| P k ˆ. Then as, by Lemma 3.15, A H is a rigid Frobenius algebra, Rep loc ZpVect ω G q pA H q is a ribbon category by [KO02,LW22a]. To check compatibility with the twist, recall that I is a ribbon functor to ZpVect ω G q, see Proposition 3.10.…”

Section: Local Modules Over Coset Algebrasmentioning

confidence: 99%

“…Given a commutative algebra A in a braided tensor category C one defines a braided tensor category Rep loc C pAq of local modules [Par95,Sch01,KO02,LW22a]. Such categories of local modules have been of particular interest in the mathematical physics literature, see e.g.…”

mentioning

confidence: 99%

“…For instance, categories of local modules relate the representations of a VOA to those of its extensions [KO02,HKL15,CKM17]. Given a rigid Frobenius algebra (i.e., a connected commutative special Frobenius algebra) in C, it was shown that the rigid monoidal category Rep loc C pAq of local modules is again modular (see [KO02] in the semisimple case, and [LW22a] in the general case). Such rigid Frobenius algebras were classified in the semisimplification of U q psl 2 q-modules [KO02], the Drinfeld center of a finite group [Dav10,LW22a], and in DW categories ZpVect ω kG q for char k " 0 in [DS17].…”

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confidence: 99%

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Frobenius monoidal functors of Dijkgraaf-Witten categories and rigid Frobenius algebras

Hannah¹,

Laugwitz²,

Camacho³

2023

Preprint

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We construct a separable Frobenius monoidal functor from ZpVect ω| H H q to ZpVect ω G q for any subgroup H of G which preserves braiding and ribbon structure. As an application, we classify rigid Frobenius algebras in ZpVect ω G q, recovering the classification of étale algebras in these categories by Davydov-Simmons [J. Algebra 471 (2017)] and generalizing their classification to algebraically closed fields of arbitrary characteristic. Categories of local modules over such algebras are modular tensor categories by results of Kirillov-Ostrik [Adv. Math 171 (2002)] in the semisimple case and Laugwitz-Walton [IMRN 2022, Issue 20 (2022] in the general case.

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Hopf algebraic methods in finite tensor categories: Characters and integrals

Shimizu¹

2021

Contemporary Mathematics

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We introduce the notion of a quasi-Frobenius algebra in a finite tensor category C and give equivalent conditions for an algebra in C to be quasi-Frobenius. A quasi-Frobenius algebra in C is not necessarily Frobenius, however, we show that an algebra A in C is quasi-Frobenius if and only if A is Morita equivalent to a Frobenius algebra in C. We also show that the class of symmetric Frobenius algebras in C is closed under the Morita equivalence provided that C is pivotal so that the symmetricity makes sense.

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Constructing non-semisimple modular categories with local modules

Cited by 3 publications

References 30 publications

A Kazhdan–Lusztig Correspondence for $$L_{-\frac{3}{2}}(\mathfrak {sl}_3)$$

A Kazhdan–Lusztig Correspondence for $$L_{-\frac{3}{2}}(\mathfrak {sl}_3)$$

Frobenius monoidal functors of Dijkgraaf-Witten categories and rigid Frobenius algebras

Hopf algebraic methods in finite tensor categories: Characters and integrals

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