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SUMMARY:Shift of argument subalgebras in Poisson algebras and their quanti
 zation. 
DTSTART:20150303T151500
DTEND:20150303T170000
DTSTAMP:20260916T085830Z
UID:bc1f4bf9a4d108bd4377f0c50c123987d29a94ea1ed8d65137476637
CATEGORIES:Conferences - Seminars
DESCRIPTION:Leonid Rybnikov\, HSE Moscow\nThe symmetric algebra S(g) of a 
 Lie algebra g carries a natural\nPoisson bracket. Shift of argument subalg
 ebras (introduced by Fomenko\nand Mishchenko in 1978) form a family of max
 imal Poisson-commutative\nsubalgebras in S(g) for semisimple g. This famil
 y is parametrized by\nregular elements of the dual space g*. I will discus
 s the quantization\nproblem for shift of argument subalgebras\, namely\, h
 ow to lift these\nsubalgebras to commutative subalgebras in the universal 
 enveloping\nalgebra U(g)\, and how to describe the spectra of the "quantum
  shift of\nargument subalgebras" of U(g) on (finite-dimensional) g-modules
 . These\nquestions are related to the classical representation theory\, in
 \nparticular\, it was observed by Vinberg\, that the Gelfand-Tsetlin\nsuba
 lgebra in U(gl_n) is a certain limit of quantum shift of argument\nsubalge
 bras. Hence the spectra of quantum shift of argument\nsubalgebras on a fin
 ite-dimensional gl_n-module can be regarded as a\ndeformation of the corre
 sponding Gelfand-Tsetlin polytope.\nThe construction of the quantum shift 
 of argument subalgebras is a\nversion of the Feigin-Frenkel-Reshetikhin co
 nstruction of higher\nhamiltonians for the Gaudin model. The\nquantum shif
 t of argument subalgebras come from the center of the\nuniversal envelopin
 g algebra of the corresponding affine Lie algebra\ng^ at the critical leve
 l  by an appropriate quantum Hamiltonian\nreduction. The center at the cr
 itical level is naturally identified\nwith the algebra of polynomial funct
 ions on the space of opers on the\nformal punctured disk with respect to t
 he Langlands dual group G^L\n(roughly\, the space of gauge equivalence cla
 sses of connections in a\nprincipal G^L-bundle with some transversality co
 ndition). This allows\nus to treat the spectra of the quantum shift of arg
 ument subalgebras\non g-modules as some subsets in the space of opers. I w
 ill give a\nprecise description of these subsets for irreducible\nfinite-d
 imensional g-modules.
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STATUS:CONFIRMED
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