Algebras of Non-Local Screenings and Diagonal Nichols Algebras

Flandoli, I.; Lentner, Simon

doi:10.3842/sigma.2022.018

Cited by 1 publication

(2 citation statements)

References 72 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the context of vertex algebras, the non-local screening operators are described by Nichols algebras in the category of representations of the vertex algebra [Len21]. For an introduction to Nichols algebras and an introduction to screening operators, see for example [FL22] Section 2 and 3.…”

Section: Definition and Relevant Examplesmentioning

confidence: 99%

See 1 more Smart Citation

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

Creutzig

Ridout

Rupert

2023

Commun. Math. Phys.

View full text Add to dashboard Cite

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.

show abstract

Section: Definition and Relevant Examplesmentioning

confidence: 99%

“…Under certain convergence conditions, the algebra of screenings is the Nichols algebra [Len21]. The question which lattices realize C in a way compatible with reflection operators was treated in [FL22], which also addreses the questions for which choices the convergence conditions holds. Problem 10.5.…”

Section: Fixing the Twistmentioning

confidence: 99%