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Let $\mathfrak g$ be a semi-simple Lie algebra. Kazhdan and Lusztig studied the category of representations of the corresponding affine Lie algebra (the central extension of $\mathfrak g((t))$) which are integrable under $\mathfrak g[[t]]$; under some assumptions on the central charge they constructed a tensor structure there and proved that the resulting tensor category is equivalent to the category of finite-dimensional representations of the corresponding quantum group (where the parameter $q$ in the quantum group depends on the central charge).

My question is the following: is there a known analog of these results for Lie super-algebras? I am particularly interested in $\mathfrak g=\text{gl}(m|n)$. Have people studied the Kazhdan-Lusztig category for Lie super-algebras?

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In: Kazhdan-Lustzig polynomials and character formulae for the Lie superalgebra $gl(m|n)$, J. Amer. Math. Soc. 16 (2002), 185–231, J. Brundan develops a conjecture on the characters for the irreducible modules and tilting modules in the full BGG category $\mathcal{O}^{m|n}$ of $gl(m|n)$ modules.

In: The Brundan–Kazhdan–Lusztig conjecture for general linear Lie superalgebras, Duke Math. J. Volume 164, Number 4 (2015), 617-695 and in:
Tensor Product Categorifications and the Super Kazhdan–Lusztig Conjecture, , IMRN, Volume 2017, Issue 20, 1 October 2017, Pages 6329–6410, independent proofs are provided.
(see also: this paper)

(The arXiv versions are: arXiv:1203.0092 [math.RT] and arXiv:1310.0349v3 [math.RT] correspondingly).

I am not a specialist to say more, but i think the results in these papers may be related to what you are looking for.

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  • $\begingroup$ Thank you. It is indeed related although not exactly the same $\endgroup$ – Alexander Braverman Feb 5 '19 at 4:21

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