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A conditional algebraic proof of the logarithmic Kazhdan-Lusztig correspondence

Simon D. Lentner

TL;DR

This work develops a conditional algebraic framework for the logarithmic Kazhdan-Lusztig correspondence by reconstructing the ambient braided tensor category from a known semisimple subcategory through Ext^1 data and Nichols algebras. Central to the approach are the relative Drinfeld center and Schauenburg’s functor, which realize the unknown category as a center of a module category over a Nichols algebra, effectively linking VOAs defined as kernels of screenings to generalized small quantum groups. The authors prove a conditional KL equivalence for Feigin-Tipunin-type algebras by matching Frobenius-Perron dimensions and using asymptotics of VOA characters, assuming $C_2$-cofiniteness and a coherence between quantum and categorical dimensions. The results unify algebraic reconstruction with geometric-analytic character methods, outlining explicit strategies to produce Ext^1 from screening data and to identify the Nichols algebra governing the underlying generalized quantum group. While providing powerful conditional results and clarifying structural obstacles with counterexamples, the paper also sketches a path to full KL equivalence once the vertex-algebra side satisfies the required analytic finiteness and rigidity properties.

Abstract

The logarithmic Kazhdan-Lusztig correspondence is a conjectural equivalence between braided tensor categories of representations of small quantum groups and representations of certain vertex operator algebras. In this article we prove such an equivalence, and more general versions, using mainly algebraic arguments that characterize the representation category of the quantum group by quantities that are accessible on the vertex algebra side. Our proof is conditional on suitable analytic properties of the vertex algebra and its representation category. More precisely, we assume that it is a finite braided rigid monoidal category where the Frobenius-Perron dimensions are given by asymptotics of analytic characters.

A conditional algebraic proof of the logarithmic Kazhdan-Lusztig correspondence

TL;DR

This work develops a conditional algebraic framework for the logarithmic Kazhdan-Lusztig correspondence by reconstructing the ambient braided tensor category from a known semisimple subcategory through Ext^1 data and Nichols algebras. Central to the approach are the relative Drinfeld center and Schauenburg’s functor, which realize the unknown category as a center of a module category over a Nichols algebra, effectively linking VOAs defined as kernels of screenings to generalized small quantum groups. The authors prove a conditional KL equivalence for Feigin-Tipunin-type algebras by matching Frobenius-Perron dimensions and using asymptotics of VOA characters, assuming -cofiniteness and a coherence between quantum and categorical dimensions. The results unify algebraic reconstruction with geometric-analytic character methods, outlining explicit strategies to produce Ext^1 from screening data and to identify the Nichols algebra governing the underlying generalized quantum group. While providing powerful conditional results and clarifying structural obstacles with counterexamples, the paper also sketches a path to full KL equivalence once the vertex-algebra side satisfies the required analytic finiteness and rigidity properties.

Abstract

The logarithmic Kazhdan-Lusztig correspondence is a conjectural equivalence between braided tensor categories of representations of small quantum groups and representations of certain vertex operator algebras. In this article we prove such an equivalence, and more general versions, using mainly algebraic arguments that characterize the representation category of the quantum group by quantities that are accessible on the vertex algebra side. Our proof is conditional on suitable analytic properties of the vertex algebra and its representation category. More precisely, we assume that it is a finite braided rigid monoidal category where the Frobenius-Perron dimensions are given by asymptotics of analytic characters.
Paper Structure (31 sections, 42 theorems, 72 equations)

This paper contains 31 sections, 42 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Conventions
  3. Background
  4. Content of this article
  5. Categorical preliminaries
  6. Braided monoidal categories
  7. Commutative algebras
  8. Simple current extensions
  9. Central functors and algebras
  10. Frobenius-Perron dimension
  11. Relative Drinfeld centers and reconstruction results
  12. Relative Drinfeld center
  13. Schauenburg functor
  14. Adjoint algebra
  15. Generalized quantum groups
  16. ...and 16 more sections

Key Result

Lemma 1

Let $\mathcal{C}$ be a braided tensor category, which is a central full abelian subcategory of a tensor category $\mathcal{B}$, then there exists a Hopf algebra $\mathfrak{N}$ in $\mathcal{C}$ and an equivalence of monoidal categories Here, the functor $\mathfrak{l}:\mathcal{B}^{\mathrm{split}}\to \mathcal{C}$ corresponds to the functor forgetting the coaction of $\mathfrak{N}$ and the full embed

Theorems & Definitions (80)

  • Lemma 1: \ref{['lm_splitLM']}
  • Example 1.2
  • Lemma 2: \ref{['lm_CoExt']}
  • Theorem 1: \ref{['thm_ReconstructNichols2']}
  • Theorem 1.3
  • Theorem 2: \ref{['thm_KLviaFP']}
  • Corollary 1.4
  • Example 2.1: Quadratic Space
  • Definition 2.2
  • Definition 2.3
  • ...and 70 more