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Quantized semisimple Lie groups

Rita Fioresi, Robert Yuncken

Abstract

These notes present a quick introduction to the q-deformations of semisimple Lie groups from the point of view of unitary representation theory. In order to remain concrete, we concentrate entirely on the case of the lie algebra $\mathrm{sl}(2,\mathbb{C})$ and its associated compact and complex semisimple Lie groups $\mathrm{SU}(2)$ and $\mathrm{SL}(2,\mathbb{C})$. We treat the following topics: The quantized enveloping algebra and its representations; Hopf algebras and the various notions of quantum groups; real structures; quantized algebras of functions on a compact semisimple group; quantized convolution algebras; the Peter-Weyl theorem; quantized complex semisimple Lie groups as quantum doubles; representations of quantized complex semisimple Lie groups; the quantum analogue of Harish-Chandra's Plancherel formula.

Quantized semisimple Lie groups

Abstract

These notes present a quick introduction to the q-deformations of semisimple Lie groups from the point of view of unitary representation theory. In order to remain concrete, we concentrate entirely on the case of the lie algebra and its associated compact and complex semisimple Lie groups and . We treat the following topics: The quantized enveloping algebra and its representations; Hopf algebras and the various notions of quantum groups; real structures; quantized algebras of functions on a compact semisimple group; quantized convolution algebras; the Peter-Weyl theorem; quantized complex semisimple Lie groups as quantum doubles; representations of quantized complex semisimple Lie groups; the quantum analogue of Harish-Chandra's Plancherel formula.
Paper Structure (29 sections, 17 theorems, 154 equations)

This paper contains 29 sections, 17 theorems, 154 equations.

Table of Contents

  1. Introduction
  2. Quantized enveloping algebras
  3. Semisimple Lie Algebras
  4. Quantized enveloping algebras
  5. Representations of $\mathcal{U}_q(\mathfrak{g})$
  6. Hopf algebras and quantum groups
  7. Motivation: Some rough remarks on Pontryagin duality
  8. Hopf algebras
  9. Quantized enveloping algebras as quantum groups
  10. Real structures and compact quantum groups
  11. Operations on Hopf algebra representations
  12. Quantized algebras of functions
  13. The algebra of polynomial functions on $G_q$
  14. Compact Quantum Groups
  15. The convolution algebra polynomials on $K_q$
  16. ...and 14 more sections

Key Result

Theorem 2.6

Every finite dimensional integrable representation of $\mathcal{U}_q(\mathfrak{g})$ is isomorphic, for a suitable $m \in {\frac{1}{2}}\mathbf{N}$ to with for all $\mu$, with the convention $v_{\mu}=0$ if $\mu\notin\{m,m-1,\ldots,-m\}$.

Theorems & Definitions (47)

  • Example 2.1
  • Definition 2.5
  • Theorem 2.6
  • Remark 2.7
  • proof
  • Definition 3.1
  • Example 3.2
  • Example 3.3
  • Example 3.4
  • Theorem 3.5: Sklyanin
  • ...and 37 more