Quiver varieties and t-analogs of q-characters of quantum affine algebras

2001/05/22 by Hiraku Nakajima, Nakajima, Hiraku · 5 citations
Mathematics · #14D21 #14L30 #16G20 #17B37 #Advanced Combinatorial Mathematics #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #FOS: Mathematics #Quantum Algebra (math.QA) #math.AG #math.QA #msc:14D21 #msc:14L30 #msc:16G20 #msc:17B37

paper · pdf · doi:10.48550/arxiv.math/0105173

32 pages, The definition of the multiplication of the $t$--analog of the representation ring is corrected. Several ref's are added

openalex publication_date 2001/05/22 · arxiv created 2002/04/14 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let us consider a specialization of an untwisted quantum affine algebra of type ADE at a nonzero complex number, which may or may not be a root of unity. The Grothendieck ring of its finite dimensional representations has two bases, simple modules and standard modules. We identify entries of the transition matrix with special values of ``computable'' polynomials, similar to Kazhdan-Lusztig polynomials. At the same time we ``compute'' q-characters for all simple modules. The result is based on ``computations'' of Betti numbers of graded/cyclic quiver varieties. (The reason why we put `` '' will be explained in the end of the introduction.)

Cited by

• Extremal monomials of q-characters
• Representations and characters of quantum affine algebras at the crossroads between cluster categorification and quantum integrable models
• Quiver moduli and quantum loop algebras
• An introduction to representation-theoretic canonical bases of cluster algebras
• Quantum cluster algebras and representations of shifted quantum affine algebras

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