Quiver varieties and finite dimensional representations of quantum affine algebras

2000/10/02 by Hiraku Nakajima · 15 citations
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Algorithm #Artificial intelligence #Computer science #Geometry #Mathematics #Nonlinear Waves and Solitons #Pure mathematics #Quiver

paper · pdf · doi:10.1090/s0894-0347-00-00353-2

openalex publication_date 2000/10/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06

Abstract

We study finite dimensional representations of the quantum affine algebra<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper U Subscript q Baseline left-parenthesis ModifyingAbove German g With caret right-parenthesis"><mml:semantics><mml:mrow><mml:msub><mml:mrow class="MJX-TeXAtom-ORD"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="bold">U</mml:mi></mml:mrow></mml:mrow><mml:mi>q</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mover><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="fraktur">g</mml:mi></mml:mrow><mml:mo>^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:annotation encoding="application/x-tex">\mathbf U_q(\widehat \mathfrak g)</mml:annotation></mml:semantics></mml:math></inline-formula>using geometry of quiver varieties introduced by the author. As an application, we obtain character formulas expressed in terms of intersection cohomologies of quiver varieties.

Citations

• Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras
• Finite Dimensional Representations of Quantum Affine Algebras
• Quantum Affine Algebras and their Representations
• Hall algebras and quantum groups
• Instantons and affine algebras I: The Hilbert scheme and vertex operators
• Quiver varieties and Kac-Moody algebras
• MODULI OF REPRESENTATIONS OF FINITE DIMENSIONAL ALGEBRAS
• Stratified Symplectic Spaces and Reduction
• Quivers, Perverse Sheaves, and Quantized Enveloping Algebras
• Canonical Bases Arising from Quantized Enveloping Algebras

Cited by

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• The Mirkovic-Vilonen basis and Duistermaat-Heckman measures
• Graded quiver varieties, quantum cluster algebras and dual canonical basis
• Quiver varieties and the quantum Knizhnik–Zamolodchikov equation
• Higher spin \mathfrak s\mathfrak l₂R-matrix from equivariant (co)homology
• Monoidal categorification and quantum affine algebras
• Affine Kac-Moody algebras, integrable systems and their deformations
• Lusztig sheaves, decomposition rule and restriction rule
• Wall-crossings and a categorification of K-theory stable bases of the\n Springer resolution
• Q-data and Representation Theory of Untwisted Quantum Affine Algebras
• Quiver W-algebras
• Hiraku Nakajima [wikipedia]