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4 results

  • Counting Higgs bundles and type $A$ quiver bundles

  • Part of
  • Sergey Mozgovoy, Olivier Schiffmann
  • Journal:
    Compositio Mathematica / Volume 156 / Issue 4 / April 2020
    Published online by Cambridge University Press:
    27 February 2020, pp. 744-769
    Print publication:
    April 2020

  • We prove a closed formula counting semistable twisted (or meromorphic) Higgs bundles of fixed rank and degree over a smooth projective curve of genus $g$ defined over a finite field, when the twisting line bundle degree is at least $2g-2$ (this includes the case of usual Higgs bundles). This yields a closed expression for the Donaldson–Thomas invariants of the moduli spaces of twisted Higgs bundles. We similarly deal with twisted quiver sheaves of type $A$ (finite or affine), obtaining in particular a Harder–Narasimhan-type formula counting semistable $U(p,q)$-Higgs bundles over a smooth projective curve defined over a finite field.

  • TWO DESCRIPTIONS OF THE QUANTUM AFFINE ALGEBRA Uv() VIA HALL ALGEBRA APPROACH

  • IGOR BURBAN, OLIVIER SCHIFFMANN
  • Journal:
    Glasgow Mathematical Journal / Volume 54 / Issue 2 / May 2012
    Published online by Cambridge University Press:
    12 December 2011, pp. 283-307
    Print publication:
    May 2012
    • Article
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  • We compare the reduced Drinfeld doubles of the composition subalgebras of the category of representations of the Kronecker quiver and the category of coherent sheaves on ℙ1. Using this approach, we show that the Drinfeld–Beck isomorphism for the quantized enveloping algebra Uv() is a corollary of an equivalence between the derived categories Db(Rep()) and Db(Coh(ℙ1)). This technique allows to reprove several results on the integral form of Uv().

  • Homological realization of Nakajima varieties and Weyl group actions

  • Igor Frenkel, Mikhail Khovanov, Olivier Schiffmann
  • Journal:
    Compositio Mathematica / Volume 141 / Issue 6 / November 2005
    Published online by Cambridge University Press:
    04 December 2007, pp. 1479-1503
    Print publication:
    November 2005
    • Article
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  • We give a realization of Nakajima varieties and the action of the Weyl group on them using certain canonical structures of homological algebras and their natural generalization, which we develop in this paper. We consider in detail the case of an affine quiver, where we present a simple homological characterization of Nakajima varieties and its relation to moduli of sheaves on the projective plane.

  • Lectures on the dynamical Yang-Baxter Equations

  • Edited by Andrew Pressley, King's College London
  • Book:
    Quantum Groups and Lie Theory
    Published online:
    05 November 2009
    Print publication:
    17 January 2002, pp 89-129

  • Summary

    Introduction

    This paper arose from a minicourse given by the first author at MIT in the Spring of 1999, when the second author extended and improved his lecture notes of this minicourse. It contains a systematic and elementary introduction to a new area of the theory of quantum groups – the theory of the classical and quantum dynamical Yang-Baxter equations.

    The quantum dynamical Yang-Baxter equation is a generalization of the ordinary quantum Yang-Baxter equation. It first appeared in physical literature in the work of Gervais and Neveu [GN], and was first considered from a mathematical viewpoint by Felder [F], who attached to every solution of this equation a quantum group, and an interesting system of difference equations, – the quantum Knizhnik-Zamolodchikov-Bernard (qKZB) equation. Felder also considered the classical analogue of the quantum dynamical Yang-Baxter equation – the classical dynamical Yang-Baxter equation. Since then, this theory was systematically developed in many papers, some of which are listed below. By now, the theory of the classical and quantum dynamical Yang-Baxter equations and their solutions has many applications, in particular to integrable systems and representation theory. To discuss this theory and some of its applications is the goal of this paper.

    The structure of the paper is as follows.

    In Section 2 we consider the exchange construction, which is a natural construction in classical representation theory that leads one to discover the quantum dynamical Yang-Baxter equation and interesting solutions of this equation (dynamical R-matrices).