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On multiplicity-free weight modules over quantum affine algebras

Xingpeng Liu

Abstract

In this note, our goal is to construct and study the multiplicity-free weight modules of quantum affine algebras. For this, we introduce the notion of shiftability condition with respect to a symmetrizable generalized Cartan matrix, and investigate its applications on the study of quantum affine algebra structures and the realizations of the infinite-dimensional multiplicity-free weight modules. We also compute the highest $\ell$-weights of the infinite-dimensional multiplicity-free weight modules as highest $\ell$-weight modules.

On multiplicity-free weight modules over quantum affine algebras

Abstract

In this note, our goal is to construct and study the multiplicity-free weight modules of quantum affine algebras. For this, we introduce the notion of shiftability condition with respect to a symmetrizable generalized Cartan matrix, and investigate its applications on the study of quantum affine algebra structures and the realizations of the infinite-dimensional multiplicity-free weight modules. We also compute the highest -weights of the infinite-dimensional multiplicity-free weight modules as highest -weight modules.
Paper Structure (18 sections, 17 theorems, 111 equations)

This paper contains 18 sections, 17 theorems, 111 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and notations
  3. Affine Kac-Moody algebras
  4. Quantum affine algebras
  5. Highest $\ell$-weight representations with finite weight multiplicities
  6. Highest $\ell$-weight modules
  7. The classification theorem: rationality
  8. Shiftability conditions and algebra homomorphisms
  9. Shiftability conditions
  10. Quantized oscillator algebra and algebra homomorphisms
  11. Multiplicity-free weight modules
  12. Module structures on $\mathcal{A}_0$
  13. Realization of multiplicity-free weight modules
  14. Highest $\ell$-weights
  15. Multiplicity-free highest $\ell$-weight modules
  16. ...and 3 more sections

Key Result

Theorem 3.2

Let $V$ be an irreducible highest $\ell$-weight module. Then all weight spaces of $V$ are finite-dimensional if and only if its highest $\ell$-weight $\boldsymbol{f}$ belongs to $\mathcal{R}$.

Theorems & Definitions (38)

  • Definition 3.1
  • Theorem 3.2
  • proof
  • Example 4.1
  • Theorem 4.2
  • Remark 4.3
  • Lemma 4.4
  • Proposition 4.5: Hay90KOS15
  • Theorem 5.1
  • proof
  • ...and 28 more