[Paper Review] An algebraic theory for logarithmic Kazhdan-Lusztig correspondences

Thomas Creutzig, Simon Lentner+1 more|arXiv (Cornell University)|Jun 20, 2023

Algebraic structures and combinatorial modelsMathematics3 citations

TL;DR

This paper develops an algebraic framework to establish logarithmic Kazhdan-Lusztig correspondences—braided tensor equivalences between categories of representations of vertex algebras and quantum groups—by characterizing a complex braided tensor category 𝒰 as a relative Drinfeld center of a simpler category via a commutative algebra A. The key result is a systematic reduction of braided tensor equivalences to abelian equivalences when A and its module categories are well-understood, enabling proofs of new equivalences, including between the singlet vertex algebra ℳ(p) and the unrolled small quantum group of 𝔰𝔩₂ at 2p-th root of unity, and between the affine gl₁|₁ vertex algebra and its quantum group counterpart.

ABSTRACT

Let $\mathcal{U}$ be a braided tensor category, typically unknown, complicated and in particular non-semisimple. We characterize $\mathcal{U}$ under the assumption that there exists a commutative algebra $A$ in $\mathcal{U}$ with certain properties: Let $\mathcal{C}$ be the category of local $A$-modules in $\mathcal{U}$ and $\mathcal{B}$ the category of $A$-modules in $\mathcal{U}$, which are in our set-up usually much simpler categories than $\mathcal{U}$. Then we can characterize $\mathcal{U}$ as a relative Drinfeld center $\mathcal{Z}_\mathcal{C}(\mathcal{B})$ and $\mathcal{B}$ as representations of a certain Hopf algebra inside $\mathcal{C}$. In particular this allows us to reduce braided tensor equivalences to the knowledge of abelian equivalences, e.g. if we already know that $\mathcal{U}$ is abelian equivalent to the category of modules of some quantum group $U_q$ or some generalization thereof, and if $\mathcal{C}$ 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 $\mathcal{U}$ and $\mathrm{Rep}(U_q)$. 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 $\mathcal{C}$ 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 $\mathcal{M}(p)$ and the unrolled small quantum groups of $\mathfrak{sl}_2$ at $2p$-th root of unity. Another new example is the Kazhdan-Lusztig correspondence for $\mathfrak{gl}_{1|1}$.

Motivation & Objective

• To establish a general framework for proving logarithmic Kazhdan-Lusztig correspondences—braided tensor equivalences between vertex algebra and quantum group representation categories.
• To reduce the problem of constructing braided tensor equivalences to verifying abelian equivalences between simpler module categories.
• To characterize a non-semisimple braided tensor category 𝒰 as a relative Drinfeld center 𝒵_𝒞(ℬ) using a commutative algebra A in 𝒰.
• To apply the theory to prove new equivalences, including between the singlet vertex algebra ℳ(p) and the unrolled small quantum group of 𝔰𝔩₂ at 2p-th root of unity.
• To extend the theory to unrolled quantum groups and affine gl₁|₁, proving a new correspondence between its vertex algebra and quantum group representations.

Proposed method

• Characterize a braided tensor category 𝒰 as the relative Drinfeld center 𝒵_𝒞(ℬ), where 𝒞 is the category of local A-modules and ℬ is the category of A-modules in 𝒰.
• Use a commutative algebra A in 𝒰 with specific properties to induce a splitting tensor functor from 𝒰 to ℬ.
• Realize ℬ as representations of a Hopf algebra inside the braided category 𝒞, enabling reconstruction of the tensor structure.
• Apply the theory to free-field realizations of vertex algebras and Nichols algebras as the source of A and 𝒞.
• Use the criterion that an exact functor is fully faithful if it induces isomorphisms on Hom-spaces between projective and injective objects.
• Leverage the existence of projective and injective resolutions to lift abelian equivalences to braided tensor equivalences.

Experimental results

Research questions

• RQ1 Under what conditions can a braided tensor category 𝒰 be reconstructed as a relative Drinfeld center of a simpler category ℬ via a commutative algebra A in 𝒰?
• RQ2 How can braided tensor equivalences between vertex algebra and quantum group categories be reduced to abelian equivalences?
• RQ3 What role do free-field realizations and Nichols algebras play in constructing the required commutative algebra A and the category 𝒞 of local A-modules?
• RQ4 Can the theory prove the Kazhdan-Lusztig correspondence for the singlet vertex algebra ℳ(p) and the unrolled small quantum group of 𝔰𝔩₂ at 2p-th root of unity?
• RQ5 Does the theory extend to affine vertex algebras like that of gl₁|₁, yielding a new correspondence with its unrolled quantum group?

Key findings

• The paper proves a braided tensor equivalence between the category of representations of the singlet vertex algebra ℳ(p) and the unrolled small quantum group of 𝔰𝔩₂ at a 2p-th root of unity.
• A new Kazhdan-Lusztig correspondence is established between the affine vertex algebra of 𝔤𝔩₁|₁ and the unrolled quantum group of 𝔤𝔩₁|₁.
• The theory reduces the construction of braided tensor equivalences to verifying abelian equivalences between ℬ and the module category of a quantum group, provided the category 𝒞 of local A-modules is braided equivalent to graded vector spaces.
• The relative Drinfeld center construction 𝒰 ≅ 𝒵_𝒞(ℬ) is shown to recover the full braided tensor structure of 𝒰 from the simpler category ℬ and the algebra A.
• The existence of projective and injective resolutions ensures that an exact functor preserving Hom-spaces between projective/injective objects is fully faithful, enabling the lifting of abelian equivalences to braided tensor equivalences.
• The framework successfully applies to both logarithmic VOAs and quantum groups, providing a general mechanism for proving such correspondences in non-semisimple settings.