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On De Concini-Kac forms of quantum groups

Ivan Losev, Alexander Tsymbaliuk, Trung Vu

Abstract

Quantum groups of semisimple Lie algebras at roots of unity admit several different forms. Among them is the De Concini-Kac form, which is the easiest to define but, perhaps, hardest to study. In this paper, we propose a suitable modification to the De Concini-Kac form, namely the even part algebra, which has some appealing features. Notably, it behaves uniformly with respect to the order of the roots of unity and admits an adjoint action of the Lusztig form. We revisit several results due to De Concini-Kac-Procesi and Tanisaki for the even part algebra. Namely, we give conceptual definitions of the Frobenius and Harish-Chandra centers and describe the entire center in terms of these two subalgebras getting a complete quantum analog of the Veldkamp theorem on the center of the universal enveloping algebras in positive characteristic. We investigate the Azumaya locus of the even part algebra over its center. We also show that the locally finite part of the even part algebra under the adjoint action of the Lusztig form is isomorphic to the reflection equation algebra, which is the quantized coordinate algebra with the product twisted by $R$-matrix. Some results on Lusztig forms at roots of unity are revisited and proved in greater generality including Kempf vanishing theorem and good filtrations on the quantized coordinate algebra.

On De Concini-Kac forms of quantum groups

Abstract

Quantum groups of semisimple Lie algebras at roots of unity admit several different forms. Among them is the De Concini-Kac form, which is the easiest to define but, perhaps, hardest to study. In this paper, we propose a suitable modification to the De Concini-Kac form, namely the even part algebra, which has some appealing features. Notably, it behaves uniformly with respect to the order of the roots of unity and admits an adjoint action of the Lusztig form. We revisit several results due to De Concini-Kac-Procesi and Tanisaki for the even part algebra. Namely, we give conceptual definitions of the Frobenius and Harish-Chandra centers and describe the entire center in terms of these two subalgebras getting a complete quantum analog of the Veldkamp theorem on the center of the universal enveloping algebras in positive characteristic. We investigate the Azumaya locus of the even part algebra over its center. We also show that the locally finite part of the even part algebra under the adjoint action of the Lusztig form is isomorphic to the reflection equation algebra, which is the quantized coordinate algebra with the product twisted by -matrix. Some results on Lusztig forms at roots of unity are revisited and proved in greater generality including Kempf vanishing theorem and good filtrations on the quantized coordinate algebra.
Paper Structure (73 sections, 164 theorems, 574 equations)

This paper contains 73 sections, 164 theorems, 574 equations.

Table of Contents

  1. Introduction
  2. Quantum groups
  3. Roots of unity case
  4. Main results
  5. Outline
  6. Acknowledgment
  7. Setting
  8. The Drinfeld-Jimbo quantum group ${\mathbf{U}}(\mathfrak{g})$
  9. The Lusztig form
  10. The De Concini-Kac form
  11. Twisted coproduct and integral forms
  12. Twisting
  13. Twisted Hopf algebra structure on ${\mathbf{U}}(\mathfrak{g})$
  14. Special choice of $\Phi$
  15. The (twisted) Lusztig form $\dot{U}_q(\mathfrak{g})$
  16. ...and 58 more sections

Key Result

Lemma 2.1

(a) (Triangular decomposition of ${\mathbf{U}}(\mathfrak{g})$) The multiplication map is an isomorphism of ${\mathbb{Q}}(v)$-vector spaces. (b) The subalgebras ${\mathbf{U}}^<, {\mathbf{U}}^>, {\mathbf{U}}^0$ are isomorphic to the algebras generated by $\{F_i\}_{i=1}^r$, $\{E_i\}_{i=1}^r$, and $\{K^\mu\}_{\mu\in Q}$, with the defining relations (DJ eqn 5), (DJ eqn 4), and (DJ eqn 1),

Theorems & Definitions (339)

  • Remark 1.1
  • Lemma 2.1
  • Remark 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Remark 2.5
  • Lemma 2.6
  • proof
  • Proposition 3.1
  • Remark 3.2
  • ...and 329 more