nLab Kazhdan-Luzstig correspondence

Idea

Kazhdan–Lusztig correspondence is an equivalence of braided monoidal categories between certain category of representations of a (version of) quantum group at a primitive root of unity and certain category of representations of an affine Lie algebra (or loop group, or a vertex operator algebra) at the level related to the degree of the root.

Literature

• George Lusztig, David Kazhdan, Tensor structures arising from affine Lie algebras, J. Amer. Math. Soc. 6 (1993) 905-947 doi, Part II: J. Amer. Math. Soc. (1993), 949-1011 doi; Part III: J. Amer. Math. Soc. (1994) 335-381 doi; Part IV: J. Amer. Math. Soc. (1994) 383-453 doi

Logarithmic Kazhdan–Lusztig

• B. L. Feigin, A. M. Gainutdinov, A. M. Semikhatov, I. Yu. Tipunin, Kazhdan–Lusztig correspondence for the representation category of the triplet W-algebra in logarithmic CFT, Theor.Math.Phys. 148 (2006) 1210–1235; Teor.Mat.Fiz. 148 (2006) 398–427 arXiv

• B. L. Feigin, A. M. Gainutdinov, A. M. Semikhatov, I. Yu. Tipunin, Kazhdan–Lusztig-dual quantum group for logarithimic extensions of Virasoro minimal models, J. Math. Phys. 48, 032303 (2007) doi

• Boris L. Feigin, Simon D. Lentner, Vertex algebras with big centre and a Kazhdan–Lusztig correspondence, Advances in Mathematics 457 (2024) 109904 doi

• Simon D. Lentner, A conditional algebraic proof of the logarithmic Kazhdan–Lusztig correspondence, arxiv:2501.10735

Other versions

• Chia-Cheng Liu; Semi-infinite cohomology and Kazhdan–Lusztig equivalence at positive level, arXiv:1807.01773

A positive level Kazhdan-Lusztig functor is defined using Arkhipov-Gaitsgory duality for affine Lie algebras. The functor sends objects in the DG category of G(O)-equivariant positive level affine Lie algebra modules to objects in the DG category of modules over Lusztig’s quantum group at a root of unity. We prove that the semi-infinite cohomology functor for positive level modules factors through the Kazhdan-Lusztig functor at positive level and the quantum group cohomology functor with respect to the positive part of Lusztig’s quantum group.

(cf. Liu’s thesis, Semi-infinite cohomology, quantum group cohomology, and the Kazhdan-Lusztig equivalence, link)