{"paper":{"title":"Poisson Geometry of Monic Matrix Polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP","math.QA","math.RT"],"primary_cat":"math-ph","authors_text":"Alexander Shapiro","submitted_at":"2014-05-15T16:52:03Z","abstract_excerpt":"We study the Poisson geometry of the first congruence subgroup $G_1[[z^{-1}]]$ 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 $G_1[[z^{-1}]]$ 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 $G"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.3909","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}