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Costantino, Francesco; Geer, Nathan; Patureau-Mirand, Bertrand

doi:10.1016/j.jpaa.2014.10.012

Cited by 52 publications

(74 citation statements)

References 27 publications

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“…A related conjectural logarithmic Kazhdan-Lusztig correspondence involves the unrolled restricted quantum group U H q (sl 2 ) at q = e πi/p and the singlet algebra W 0 A1 (p) [CGP15], the latter being the U (1)-orbifold of W A1 (p). This in turn means that W A1 (p) is a simple current extension of W 0 A1 (p) [RW14].…”

Section: Logarithmic Kazhdan-lusztig Correspondencesmentioning

confidence: 99%

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

Creutzig¹,

Ridout²,

Rupert³

2021

Preprint

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The abelian and monoidal structure of the category of smooth weight modules over a non-integrable affine vertex algebra of rank greater than one is an interesting, difficult and essentially wide open problem. Even conjectures are lacking. This work details and tests such a conjecture for L − 3 2 (sl3) via a logarithmic Kazhdan-Lusztig correspondence.We first investigate the representation theory of U H i (sl3), the unrolled restricted quantum group of sl3 at fourth root of unity. In particular, we analyse its finite-dimensional weight category, determining Loewy diagrams for all projective indecomposables and decomposing all tensor products of irreducibles. Our motivation is that this category is conjecturally braided tensor equivalent to a category of W 0A 2 (2)modules. Here, W 0 A 2 (2) is an orbifold of the octuplet vertex algebra WA 2 (2) of Semikhatov, the latter being the natural sl3-analogue of the well known triplet algebra. Moreover, W 0 A 2 (2) is the parafermionic coset of the affine vertex algebra LWe formulate an explicit conjecture relating the representation theory of W 0 A 2 (2) and U H i (sl3) and work out the resulting structures of the corresponding L − 3 2 (sl3)-modules. In particular, we obtain conjectural Loewy diagrams for the latter's projective indecomposables and decompositions for the fusion products of its irreducibles. These products coincide with those recently computed via Verlinde's formula. Finally, we give analogous results for WA 2 (2).

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Section: Logarithmic Kazhdan-lusztig Correspondencesmentioning

confidence: 99%

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

Creutzig¹,

Ridout²,

Rupert³

2021

Preprint

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“…Clearly then, when W is simple, Φ V,W = Ψ λ+1−p (χ(V )) Id W . Using the fact that the Hopf link S V,W is the trace of the open Hopf link Φ V,W and the appropriate character formulas [CGP,Equation (16)], it is easy to show that S (0,0,0),(γ 2 ,j,ℓ) = e πi(2γ 1 γ 2 +pkl) {(i + 1)(j + 1)} {j + 1} .…”

Section: Comparison For Odd Pmentioning

confidence: 99%

“…Then the following holds Proposition 6. [CGP,Theorem 5.2 and Lemma 5.3] 1. The typical V α are projective.…”

Section: Introductionmentioning

confidence: 99%

Braided Tensor Categories Related to $${\mathcal {B}}_{p}$$ Vertex Algebras

Auger

Creutzig

Kanade

et al. 2020

Commun. Math. Phys.

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The B p -algebras are a family of vertex operator algebras parameterized by p ∈ Z ≥2 . They are important examples of logarithmic CFTs and appear as chiral algebras of type (A 1 , A 2p−3 ) Argyres-Douglas theories. The first member of this series, the B 2 -algebra, are the well-known symplectic bosons also often called the βγ vertex operator algebra.We study categories related to the B p vertex operator algebras using their conjectural relation to unrolled restricted quantum groups of sl 2 . These categories are braided, rigid and non semisimple tensor categories. We list their simple and projective objects, their tensor products and their Hopf links. The latter are succesfully compared to modular data of characters thus confirming a proposed Verlinde formula of David Ridout and the second author.

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“…We consider specifically the categories rep SL(2) q , rep( uq (sl 2 )), and rep(u q (sl 2 )) of character graded representations of Lusztig's divided power algebra U Lus q (sl 2 ), character graded representations of small quantum SL (2), and usual representations of small quantum SL( 2) respectively (see Sections 3.1 and 12.1). We establish the following collection of equivalences, which were conjectured across the works [41,12,23,14,16]. Theorem (9.5/10.1/12.1).…”

mentioning

confidence: 91%

“…At the particular parameter p = 2, Creutzig, Lentner, and Rupert verified that this equivalence can in fact be enhanced with the desired tensor structure [20]. The possibility of the equivalence Ψ was alluded to in the works of Creutzig and Milas, and Costantino, Geer, and Patureau-Mirand [23,14], then was conjectured explicitly in work of Creutzig, Gainutdinov, and Runkel [16] (see also Remark 12.2).…”

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confidence: 99%

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

Gannon,

Negron

2021

Preprint

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We provide a ribbon tensor equivalence between the representation category of small quantum SL(2), at parameter q = e πi/p , and the representation category of the triplet vertex operator algebra at integral parameter p > 1. We provide similar quantum group equivalences for representation categories associated to the Virasoro, and singlet vertex operator algebras at central charge c = 1 − 6(p − 1) 2 /p. These results resolve a number of fundamental conjectures coming from studies of logarithmic CFTs in type A 1 .

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Some remarks on the unrolled quantum group of sl(2)

Cited by 52 publications

References 27 publications

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

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

Braided Tensor Categories Related to $${\mathcal {B}}_{p}$$ Vertex Algebras

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

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