Invariant prime ideals in quantizations of nilpotent Lie algebras

By M. Yakimov. 2009

68 citations recorded by the source corpus.

De Concini, Kac, and Procesi defined a family of subalgebras 𝒰w+ of a quantized universal enveloping algebra 𝒰q( g ), associated to the elements of the corresponding Weyl group W. They are deformations of the universal enveloping algebras 𝒰( n+∩ Adw( n−)), where n± are the nilradicals of a pair of dual Borel subalgebras. Based on the results of Gorelik and Joseph and an interpretation of 𝒰w+ as quantized algebras of functions on Schubert cells, we construct explicitly the H‐invariant prime ideals of each 𝒰w+ and show that the corresponding poset is isomorphic to W⩽w, where H is the group of group‐like elements of 𝒰q( g ). Moreover, for each H‐prime of 𝒰w+ we construct a generating set in terms of Demazure modules related to fundamental representations. Using the results of Ramanathan and Kempf, we prove similar theorems for vanishing ideals of closures of torus orbits of symplectic leaves of related Poisson structures on Schubert cells in flag varieties.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/eaac7fa78637c25cfa8e2cc52c89a6df065a7e11