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On Borel subalgebras of quantum groups

Simon D. Lentner, Karolina Vocke

Abstract

For a quantum group, we study those right coideal subalgebras, for which all irreducible representations are one-dimensional. If a right coideal subalgebra is maximal with this property, then we call it a Borel subalgebra. Besides the positive part of the quantum group and its reflections, we find new unfamiliar Borel subalgebras, for example, ones containing copies of the quantum Weyl algebra. Given a Borel subalgebra, we study its induced (Verma-)modules and prove among others that they have all irreducible finite-dimensional modules as quotients. We give two structural conjectures involving the associated graded right coideal subalgebra, which we prove in certain cases. In particular, they predict the shape of all triangular Borel subalgebras. As examples, we determine all Borel subalgebras of $U_q(\mathfrak{sl}_2)$ and $U_q(\mathfrak{sl}_3)$ and discuss the induced modules.

On Borel subalgebras of quantum groups

Abstract

For a quantum group, we study those right coideal subalgebras, for which all irreducible representations are one-dimensional. If a right coideal subalgebra is maximal with this property, then we call it a Borel subalgebra. Besides the positive part of the quantum group and its reflections, we find new unfamiliar Borel subalgebras, for example, ones containing copies of the quantum Weyl algebra. Given a Borel subalgebra, we study its induced (Verma-)modules and prove among others that they have all irreducible finite-dimensional modules as quotients. We give two structural conjectures involving the associated graded right coideal subalgebra, which we prove in certain cases. In particular, they predict the shape of all triangular Borel subalgebras. As examples, we determine all Borel subalgebras of and and discuss the induced modules.

Paper Structure

This paper contains 29 sections, 30 theorems, 142 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quantum groups
  4. Quantum group representation theory
  5. Borel subalgebras
  6. Main Definitions
  7. Example A1
  8. Homogeneous Borel subalgebras
  9. Induction of one-dimensional characters
  10. Definition and first properties
  11. Example A1
  12. The structure of the graded algebra of a right coideal subalgebra
  13. Conjecture A
  14. Localization
  15. Growth conditions
  16. ...and 14 more sections

Key Result

Theorem \oldthetheorem

For every $w\in W$ there is a right coideal subalgebra $U^+[w]U^0=U^0U^+[w]$. Conversely, every homogeneous right coideal subalgebra $C\subset U^{\geq 0}$ is of this form for some $w\in W$.

Figures (2)

  • Figure 1: Picture of $\Phi^+(w_\pm)$ with gray lines indicating character shifts. The contained Weyl algebra is visible on the $X$-axis.
  • Figure 2: Picture of $\Phi^+(w_\pm)$ with gray lines indicating character shifts. We have two Weyl algebras, one extending another.

Theorems & Definitions (80)

  • Example 1: \ref{['Weyl algebra']}
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Example \oldthetheorem
  • Theorem \oldthetheorem: HS09 Theorem 7.3
  • Theorem \oldthetheorem: HK11b Theorem 2.12
  • proof
  • Theorem \oldthetheorem: HK11b Theorem 2.17
  • ...and 70 more