Constructing Non-Semisimple Modular Categories With Relative Monoidal Centers

Laugwitz, Robert; Walton, Chelsea

doi:10.1093/imrn/rnab097

Cited by 4 publications

(3 citation statements)

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“…Here, the Drinfeld center Z(B) is a braided tensor category associated to any tensor category B, and there is a notion of a relative Drinfeld center Z C (B) with respect to a suitable braided subcategory C, see [LW1] and Definition 3.2 below. This functor is in fact an equivalence for finite categories, due to dimension arguments, see Corollary 3.8.…”

Section: Characterizing Braided Tensor Categoriesmentioning

confidence: 99%

“…In Section 6 we return to the category U , which can by our results so far be realized as N-N-Yetter-Drinfeld modules over C, see Definition 6.1. In the case C = Vect Q Γ and trivial associator we also give in Lemma 6.8 a concrete realization of this category as representations of a generalized quantum group U q , compare again [LW1]. For example the Nichols algebra in Example 4.9 produces the small quantum group U q = u q (sl 2 ) for C = Vect Zp for q of odd order p, respectively the quasi-Hopf algebra U q = ũq (sl 2 ) for C = Vect Z 2p for q of even order 2p [CGR20, GLO18], respectively the unrolled quantum group u H q (sl 2 ) for…”

Section: C[x] Elsementioning

confidence: 99%

“…The preceeding categories can be realized as (quasi-) Hopf algebras if C is realized by a quasi-triangular (quasi-) Hopf algebra C. This appears in different language and level of generality since the beginning of quantum groups, see [Lusz93,Maj94,Som96] in a quantum group setting, [HY10,AY13] in a Nichols algebra setting and [Lau20,LW1] in a terms of relative Drinfeld centers, and notably example of a Lie superalgebra in [SL23]. If the base category has a nontrivial associator category one can similarly obtain a quasi-Hopf algebra, see [GLO18] Section 5.5.…”

Section: Realizing Hopf Algebrasmentioning

confidence: 99%

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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: Characterizing Braided Tensor Categoriesmentioning

confidence: 99%

Section: C[x] Elsementioning

confidence: 99%