Yangians and cohomology rings of Laumon spaces |
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Authors: | Boris Feigin Michael Finkelberg Andrei Negut Leonid Rybnikov |
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Institution: | 1. Landau Institute for Theoretical Physics, Kosygina st 2, 117940, Moscow, Russia 2. Department of Mathematics, IMU, IITP, and State University Higher School of Economics, 20 Myasnitskaya st, 101000, Moscow, Russia 3. Simion Stoilow Institute of Mathematics of the Romanian Academy, Calea Grivitei nr. 21, 010702, Bucuresti, Romania 4. Department of Mathematics, Institute for the Information Transmission Problems and State University Higher School of Economics, 20 Myasnitskaya st, 101000, Moscow, Russia
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Abstract: | Laumon moduli spaces are certain smooth closures of the moduli spaces of maps from the projective line to the flag variety
of GL
n
. We construct the action of the Yangian of
\mathfraksln{\mathfrak{sl}_n} in the cohomology of Laumon spaces by certain natural correspondences. We construct the action of the affine Yangian (two-parametric
deformation of the universal enveloping algebra of the universal central extension of
\mathfrakslns±1,t]{\mathfrak{sl}_ns^{\pm1},t]}) in the cohomology of the affine version of Laumon spaces. We compute the matrix coefficients of the generators of the affine
Yangian in the fixed point basis of cohomology. This basis is an affine analog of the Gelfand-Tsetlin basis. The affine analog
of the Gelfand-Tsetlin algebra surjects onto the equivariant cohomology rings of the affine Laumon spaces. The cohomology
ring of the moduli space
\mathfrakMn,d{\mathfrak{M}_{n,d}} of torsion free sheaves on the plane, of rank n and second Chern class d, trivialized at infinity, is naturally embedded into the cohomology ring of certain affine Laumon space. It is the image
of the center Z of the Yangian of
\mathfrakgln{\mathfrak{gl}_n} naturally embedded into the affine Yangian. In particular, the first Chern class of the determinant line bundle on
\mathfrakMn,d{\mathfrak{M}_{n,d}} is the image of a noncommutative power sum in Z. |
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