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The Kazhdan-Lusztig correspondence, a significant concept in representation theory, is gaining increased attention. This correspondence establishes an equivalence between representation categories of quantum groups and affine Lie algebras. Recent research explores its applications in areas like logarithmic conformal field theory (CFT), particularly concerning the representation category of the triplet W-algebra. The Kazhdan-Lusztig correspondence has also been investigated in relation to vertex algebras, further solidifying its importance across different mathematical and physical domains.
Dennis Gaitsgory was awarded the Breakthrough Prize in Mathematics for his broad contributions to the field, including work closely related to representation theory and the geometric Langlands program. His recognition highlights the impact of representation theory on other areas of mathematics. Further research is focusing on exploring tensor structures arising from affine Lie algebras and building on Kazhdan and Lusztig's foundational work in the area.
Recent work has also explored the Kazhdan-Lusztig correspondence at a positive level using Arkhipov-Gaitsgory duality for affine Lie algebras. A functor is defined which 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. Researchers are actively working to 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.
References :
Dennis Gaitsgory was awarded the Breakthrough Prize in Mathematics for his broad contributions to the field, including work closely related to representation theory and the geometric Langlands program. His recognition highlights the impact of representation theory on other areas of mathematics. Further research is focusing on exploring tensor structures arising from affine Lie algebras and building on Kazhdan and Lusztig's foundational work in the area.
Recent work has also explored the Kazhdan-Lusztig correspondence at a positive level using Arkhipov-Gaitsgory duality for affine Lie algebras. A functor is defined which 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. Researchers are actively working to 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.
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References :
- nLab: Kazhdan-Luzstig correspondence.
Classification:
- HashTags: #RepresentationTheory #QuantumGroups #LieAlgebras
- Company: University
- Target: Representation
- Product: Mathematics
- Feature: Equivalence
- Type: Research
- Severity: Important