1995/02/20 by Victor Ginzburg, Mikhail Kapranov, Ginzburg, Victor +4
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Affine Lie algebra #Algebra over a field #Algebra representation #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #Cellular algebra #Conjecture #Current algebra #FOS: Mathematics #Hecke algebra #Langlands program #Mathematics #Pure mathematics #Quantum Algebra (math.QA) #Quantum affine algebra #Universal enveloping algebra #alg-geom #math.AG #math.QA #q-alg
paper · pdf · doi:10.48550/arxiv.q-alg/9502013
13 pages, submitted to Mathematical Research Letters
arxiv created 1995/02/20 · openalex publication_date 1995/02/20 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
This note is an attempt to extend "Geometric Langlands Conjecture" from algebraic curves to algebraic surfaces. We introduce certain Hecke-type operators on vector bundles on an algebraic surface. The crucial observation is that the algebra generated by the Hecke operators turns out to be a homomorphic image of the \it quantum toroidal algebra. The latter is a quantization, in the spirit of Drinfeld-Jimbo, of the universal enveloping algebra of the universal central extension of a "double-loop" Lie algebra. This yields, in particular, a new geometric construction of affine quantum groups of types A, D E in terms of Hecke operators for an elliptic surface.