Quantum SL(2) and logarithmic vertex operator algebras at (p,1)-central charge

Gannon, Terry; Negron, Cris

doi:10.48550/arxiv.2104.12821

Cited by 7 publications

(11 citation statements)

References 53 publications

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“…In the next section, we will need the M(p)-module inductions of simple L(p)-modules in O p ; a couple cases of the following proposition were also obtained in [GN,Lemma 11.4]: Proposition 2.9. For r ≥ 1 and 1 ≤ s ≤ p,…”

Section: Preliminariesmentioning

confidence: 99%

“…The logarithmic Kazhdan-Lusztig correspondence refers to equivalences of non-semisimple braided tensor categories associated to quantum groups and vertex operator algebras. The best-known example is the correspondence between a quasi-Hopf modification of the restricted quantum group of sl 2 at a 2pth root of unity and the triplet algebra W(p) [FGST1,FGST2,FHST,NT,CGR,CLR,GN]. But there is also a conjectural correspondence between our O T M(p) and the category of finite-dimensional weight modules for the unrolled restricted quantum group of sl 2 at a 2pth root of unity [CGP2,CMR], so far proved only for the atypical subcategories [GN].…”

Section: Introductionmentioning

confidence: 99%

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Ribbon tensor structure on the full representation categories of the singlet vertex algebras

Creutzig¹,

McRae²,

Yang³

2022

Preprint

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We show that the category of finite-length generalized modules for the singlet vertex algebra M(p), p ∈ Z>1, is equal to the category O M(p) of C1-cofinite M(p)modules, and that this category admits the vertex algebraic braided tensor category structure of Huang-Lepowsky-Zhang. Since O M(p) includes the uncountably many typical M(p)-modules, which are simple M(p)-module structures on Heisenberg Fock modules, our results substantially extend our previous work on tensor categories of atypical M(p)modules. We also introduce a tensor subcategory O T M(p) , graded by an algebraic torus T , which has enough projectives and is conjecturally tensor equivalent to the category of finite-dimensional weight modules for the unrolled restricted quantum group of sl2 at a 2pth root of unity. We compute all tensor products involving simple and projective M(p)modules, and we prove that both tensor categories O M(p) and O T M(p) are rigid and thus also ribbon. As an application, we use vertex operator algebra extension theory to show that the representation categories of all finite cyclic orbifolds of the triplet vertex algebras W(p) are non-semisimple modular tensor categories, and we confirm a conjecture of Adamović-Lin-Milas on the classification of simple modules for these finite cyclic orbifolds.

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Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Ribbon tensor structure on the full representation categories of the singlet vertex algebras

Creutzig¹,

McRae²,

Yang³

2022

Preprint

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“…Consider the non-local screening operators Z a i associated to inversely rescaled coroots a i = α ∨ i / √ p. Then the kernel of all Z a i defines a vertex subalgebra W ⊂ V, whose category of representations is conjecturally a non-semisimple modular tensor category equivalent to the category of representations of the small quantum group u q (g), q = e 2πi 2p , more precisely to some quasi-Hopf algebra variant. In the smallest case g = sl 2 the conjecture was solved affirmatively, after about 20 years of research by several groups [1,12,14,28,30,53,61]. For quantum groups associated to arbitrary g see [2,21,27,49].…”

Section: Introductionmentioning

confidence: 95%

Algebras of Non-Local Screenings and Diagonal Nichols Algebras

Flandoli¹,

Lentner²

2022

SIGMA

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In a vertex algebra setting, we consider non-local screening operators associated to the basis of any non-integral lattice. We have previously shown that, under certain restrictions, these screening operators satisfy the relations of a quantum shuffle algebra or Nichols algebra associated to a diagonal braiding, which encodes the non-locality and non-integrality. In the present article, we take all finite-dimensional diagonal Nichols algebras, as classified by Heckenberger, and find all lattice realizations of the braiding that are compatible with reflections. Usually, the realizations are unique or come as one- or two-parameter families. Examples include realizations of Lie superalgebras. We then study the associated algebra of screenings with improved methods. Typically, for positive definite lattices we obtain the Nichols algebra, such as the positive part of the quantum group, and for negative definite lattices we obtain a certain extension of the Nichols algebra generalizing the infinite quantum group with a large center.

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“…Thus we now review some properties of N-graded C 2 -cofinite vertex operator algebras. The following spanning set result is [Mi2,Lemma 2.4], and the W = V , w = 1 case is [GN,Proposition 8]: Lemma 2.8. Let V be an N-graded C 2 -cofinite vertex operator algebra and let T ⊆ V be a finite-dimensional graded subspace such that V = T + C 2 (V ).…”

Section: Preliminariesmentioning

confidence: 99%

“…The tensor category of grading-restricted generalized W(p)modules is rigid [TW] (see also [MY,Theorem 7.6]), so it is factorizable by Theorem 5.3. Note, however, that it is not too difficult to show directly that the braiding on the category of W(p)-modules is nondegenerate, as was observed for example in [GN,Theorem 4.7].…”

Section: Factorizability From Rigiditymentioning

confidence: 99%

On rationality for $C_2$-cofinite vertex operator algebras

McRae¹

2021

Preprint

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Let V be an N-graded, simple, self-contragredient, C2-cofinite vertex operator algebra. We show that if the S-transformation of the character of V is a linear combination of characters of V -modules, then the category C of grading-restricted generalized V -modules is a rigid tensor category. We further show, without any assumption on the character of V but assuming that C is rigid, that C is a factorizable finite ribbon category, that is, a not-necessarily-semisimple modular tensor category. As a consequence, we show that if the Zhu algebra of V is semisimple, then C is semisimple and thus V is rational. The proofs of these theorems use techniques and results from tensor categories together with the method of Moore-Seiberg and Huang for deriving identities of two-point genus-one correlation functions associated to V .We give two main applications. First, we prove the conjecture of Kac-Wakimoto and Arakawa that C2-cofinite affine W -algebras obtained via quantum Drinfeld-Sokolov reduction of admissible-level affine vertex algebras are strongly rational. The proof uses the recent result of Arakawa and van Ekeren that such W -algebras have semisimple (Ramond twisted) Zhu algebras. Second, we use our rigidity results to reduce the "coset rationality problem" to the problem of C2-cofiniteness for the coset. That is, given a vertex operator algebra inclusion U ⊗ V ֒→ A with A, U strongly rational and U , V a pair of mutual commutant subalgebras in A, we show that V is also strongly rational provided it is C2-cofinite. ROBERT MCRAE 6. Applications 61 6.1. Rationality for C 2 -cofinite W -algebras 61 6.2. Rationality for C 2 -cofinite cosets 66 Appendix A. Differential equations 69 Appendix B. The assumption in Theorem 6.4 for odd nilpotent orbits 74 B.1. Classical types 74 B.2. Exceptional types 76 References 82

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Quantum SL(2) and logarithmic vertex operator algebras at (p,1)-central charge

Cited by 7 publications

References 53 publications

Ribbon tensor structure on the full representation categories of the singlet vertex algebras

Ribbon tensor structure on the full representation categories of the singlet vertex algebras

Algebras of Non-Local Screenings and Diagonal Nichols Algebras

On rationality for $C_2$-cofinite vertex operator algebras

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