Generalized parafermions of orthogonal type

Creutzig, Thomas; Kovalchuk, V.; Linshaw, Andrew R.

doi:10.1016/j.jalgebra.2021.11.014

Cited by 5 publications

(1 citation statement)

References 23 publications

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“…(3) If g is simply-laced then O k (g) is a fusion category [15]. (4) O k (g) is a fusion category if g is of type B and if the denominator of the admissible level is 2 [21]. (5) If g is simply-laced, then a simple current modification of O k (g) is braided tensor equivalent to a subcategory of the principal W-algebra of g at level 1 − h ∨ + 1 k+h ∨ with h ∨ the dual Coxeter number of g [15].…”

Section: Introductionmentioning

confidence: 99%

Category $\mathcal O$ for vertex algebras of $\mathfrak{osp}_{1|2n}$

Creutzig¹,

Genra²,

Linshaw³

2022

Preprint

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We show that the affine vertex superalgebra V k (osp 1|2n ) at generic level k embeds in the equivariant W-algebra of sp 2n times 4n free fermions. This has two corollaries: (1) it provides a new proof that for generic k, the coset Com(2k+2n+1 , and ( 2) we obtain the decomposition of ordinary V k (osp 1|2n )-modules into V k (sp 2n ) ⊗ W ℓ (sp 2n )-modules. Next, if k is an admissible level and ℓ is a non-degenerate admissible level for sp 2n , we show that the simple algebra L k (osp 1|2n ) is an extension of the simple subalgebra L k (sp 2n )⊗W ℓ (sp 2n ). Using the theory of vertex superalgebra extensions, we prove that the category of ordinary L k (osp 1|2n )-modules is a semisimple, rigid vertex tensor supercategory with only finitely many inequivalent simple objects. It is equivalent to a certain subcategory of W ℓ (sp 2n )modules. A similar result also holds for the category of Ramond twisted modules. Due to a recent theorem of Robert McRae, we get as a corollary that categories of ordinary L k (sp 2n )-modules are rigid.

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Section: Introductionmentioning

confidence: 99%

Category $\mathcal O$ for vertex algebras of $\mathfrak{osp}_{1|2n}$

Creutzig¹,

Genra²,

Linshaw³

2022

Preprint

Self Cite

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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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A general mirror equivalence theorem for coset vertex operator algebras

McRae

2024

Sci. China Math.

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Generalized parafermions of orthogonal type

Cited by 5 publications

References 23 publications

Category $\mathcal O$ for vertex algebras of $\mathfrak{osp}_{1|2n}$

Category $\mathcal O$ for vertex algebras of $\mathfrak{osp}_{1|2n}$

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

A general mirror equivalence theorem for coset vertex operator algebras

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