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

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 $$\mathrm{Spec} R_q[G]$$.

• Primitive ideals of $$R_q[G]$$ and a Dixmier map for $$R_q[G]$$
• Separation of variables for the algebras $$S^\pm_w$$
• 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