Poisson Geometry of Monic Matrix Polynomials

2014/05/15 by Shapiro, Alexander
#FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Algebra (math.QA) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1405.3909

Abstract

We study the Poisson geometry of the first congruence subgroup G₁[[z^-1]] of the loop group G[[z^-1]] endowed with the rational r-matrix Poisson structure for G=GL_m and SL_m. We classify all the symplectic leaves on a certain ind-subvariety of G₁[[z^-1]] in terms of Smith Normal Forms. This classification extends known descriptions of symplectic leaves on the (thin) affine Grassmannian and the space of SL_m-monopoles. We show that a generic leaf is covered by open charts with Poisson transition functions, the charts being birationally isomorphic to products of coadjoint GL_m orbits. Finally, we discuss our results in terms of (thick) affine Grassmannians and Zastava spaces.

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