Categorification of integrable representations of quantum groups

Abstract

We categorify the highest weight integrable representations and their tensor products of a symmetric quantum Kac-Moody algebra. As byproducts, we obtain a geometric realization of Lusztig's canonical bases of these representations as well as a new positivity result. The main ingredient in the underlying geometric construction is a class of micro-local perverse sheaves on quiver varieties.

35pages, v2: minor revisions, typo fixed

References in corpus (4)

• Category O and sl(k) link invariants
• Microlocal Perverse Sheaves
• A geometric categorification of tensor products of $U_q(s{l}_2)$-modules
• Positivity and the canonical basis of tensor products of finite-dimensional irreducible representations of quantum sl(k)

Cited by in corpus (15)

• A diagrammatic approach to categorification of quantum groups I
• A diagrammatic approach to categorification of quantum groups III
• Knot invariants and higher representation theory
• Knot invariants and higher representation theory I: diagrammatic and geometric categorification of tensor products
• An approach to categorification of some small quantum groups II
• Brundan-Kazhdan-Lusztig conjecture for general linear Lie superalgebras
• An approach to categorification of some small quantum groups
• Lectures on canonical and crystal bases of Hall algebras
• A categorical action on quantized quiver varieties
• An approach to categorification of Verma modules
• Quiver Hecke algebras and 2-Lie algebras
• Compatibility of t-structures for quantum symplectic resolutions
• An equivalence between truncations of categorified quantum groups and Heisenberg categories
• Associated graded of Hodge modules and categorical sl_2 actions
• Categorification of Lie algebras [d'apres Rouquier, Khovanov-Lauda]