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On the Spectra of Quantum Groups
Milen Yakimov, Louisiana State University, Baton Rouge, Louisiana

Memoirs of the American Mathematical Society
2013; 91 pp; softcover
Volume: 229
ISBN-10: 0-8218-9174-X
ISBN-13: 978-0-8218-9174-2
List Price: US$71
Individual Members: US$42.60
Institutional Members: US$56.80
Order Code: MEMO/229/1078
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Joseph and Hodges-Levasseur (in the A case) described the spectra of all quantum function algebras \(R_q[G]\) on simple algebraic groups in terms of the centers of certain localizations of quotients of \(R_q[G]\) by torus invariant prime ideals, or equivalently in terms of orbits of finite groups. These centers were only known up to finite extensions. The author determines the centers explicitly under the general conditions that the deformation parameter is not a root of unity and without any restriction on the characteristic of the ground field. From it he deduces a more explicit description of all prime ideals of \(R_q[G]\) than the previously known ones and an explicit parametrization of \(\mathrm{Spec} R_q[G]\).

Table of Contents

  • Introduction
  • Previous results on spectra of quantum function algebras
  • A description of the centers of Joseph's localizations
  • Primitive ideals of \(R_q[G]\) and a Dixmier map for \(R_q[G]\)
  • Separation of variables for the algebras \(S^\pm_w\)
  • A classification of the normal and prime elements of the De Concini-Kac-Procesi algebras
  • Module structure of \(R_{\mathbf{w}}\) over their subalgebras generated by Joseph's normal elements
  • A classification of maximal ideals of \(R_q[G]\) and a question of Goodearl and Zhang
  • Chain properties and homological applications
  • Bibliography
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