Prime and Primitive Ideals in Graded Deformations of Algebraic Quantum Groups at Roots of Unity

Abstract

Let G be a connected, simply connected complex semisimple Lie group of rank n. The deformations employed by Artin, Schelter and Tate, and Hodges, Levasseur and Toro can be applied to the single parameter quantizations, at roots of unity, of the Hopf algebra of regular functions on G. Each of the resulting complex multiparameter quantum groups F∈,p[G] depends on both a suitable root of unity ∈ and an antisymmetric bicharacter p: Z n ×Z n →C×. These quantizations differ significantly from their single parameter (root-of-unity) counterparts, and, in particular, may have infinite-dimensional irreducible representations. Our approach to F∈,p[G] depends on a natural ℋ×ℋ-action thereon, where ℋ is an n-torus, and our main result offers a classification of the primitive ideals: We use a multiparameter quantum Frobenius map to provide a bijection from (Prim F∈,p[G])/ℋ×ℋ onto G/H×H, where H is a maximal torus of G. In the single parameter case, this bijection is a consequence of work by De Concini and Lyubashenko, and De Concini and Procesi; our results require their analysis. Our methods also exploit earlier work by Moeglin and Rentschler concerning actions of algebraic groups on complex Noetherian algebras. In contrast to generic quantizations of the coordinate ring of G, the primitive spectrum of F∈,p[G] is not finitely stratified by the torus action.