Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Central elements of the degenerate quantum general linear group

Hengyun Yang, Yang Zhang

TL;DR

The paper constructs central elements for the degenerate quantum general linear group $U_q( rak{gl}_{m,n})$, including an explicit quantum Casimir, by leveraging explicit $L$-operators and a partial-trace centrality framework. It introduces a spectral-parameter universal $L$-operator $L(x)$ that satisfies the quantum Yang–Baxter equation with the fundamental $R$-matrix, enabling an $RLL$ realisation of the algebra. The authors also develop an explicit $RLL$ realisation via Hopf-algebra isomorphisms between the standard degenerate quantum group and a universal RTT algebra, providing concrete mappings and a detailed ${ m U}_q( rak{gl}_{1,1})$ example. These results connect degenerate quantum groups to integrable systems and offer tools for studying their representation theory and potential lattice models.

Abstract

We construct central elements of the degenerate quantum general linear group introduced by Cheng, Wang and Zhang. In particular, we give an explicit formula for the quantum Casimir element. Our method is based on the explicit $L$ operators. Moreover, we construct a universal $L$ operator, which is a spectral parameter-dependent solution of the quantum Yang-Baxter equation in the tensor product of the degenerate quantum general linear group and the endomorphism ring of its natural representation. This construction leads to the FRT approach to the degenerate quantum general linear group.

Central elements of the degenerate quantum general linear group

TL;DR

The paper constructs central elements for the degenerate quantum general linear group , including an explicit quantum Casimir, by leveraging explicit -operators and a partial-trace centrality framework. It introduces a spectral-parameter universal -operator that satisfies the quantum Yang–Baxter equation with the fundamental -matrix, enabling an realisation of the algebra. The authors also develop an explicit realisation via Hopf-algebra isomorphisms between the standard degenerate quantum group and a universal RTT algebra, providing concrete mappings and a detailed example. These results connect degenerate quantum groups to integrable systems and offer tools for studying their representation theory and potential lattice models.

Abstract

We construct central elements of the degenerate quantum general linear group introduced by Cheng, Wang and Zhang. In particular, we give an explicit formula for the quantum Casimir element. Our method is based on the explicit operators. Moreover, we construct a universal operator, which is a spectral parameter-dependent solution of the quantum Yang-Baxter equation in the tensor product of the degenerate quantum general linear group and the endomorphism ring of its natural representation. This construction leads to the FRT approach to the degenerate quantum general linear group.
Paper Structure (11 sections, 14 theorems, 110 equations)

This paper contains 11 sections, 14 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. The degenerate quantum general linear group
  3. The degenerate quantum general linear group
  4. Commutation relations
  5. Central elements of Uq(glmn)
  6. A general construction
  7. Explicit formulae for central elements
  8. A universal L-operator
  9. The main result
  10. Proof of Theorem \ref{['thm-1']}
  11. The RLL realisation

Key Result

Lemma 2.1

For any $a\neq b\in {\bf I}_{m,n}$, the following relations hold in ${\rm U}_{q}(\mathfrak {gl}_{m,n})$: These relations similarly apply to $\bar{E}_{ab}$ mutatis mutandis.

Theorems & Definitions (30)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Theorem 3.1
  • proof
  • ...and 20 more