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Representations of Quantum Affine General Linear Superalgebras at Arbitrary 01-Sequences

Hongda Lin, Honglian Zhang

TL;DR

This work develops a comprehensive RTT-based framework for quantum affine general linear superalgebras with arbitrary 01-sequences, establishing an explicit PBW basis and transferring finite-dimensional representation theory from the standard to nonstandard parities via odd reflections. It provides necessary and sufficient conditions for the finite-dimensionality of highest-weight modules, introduces evaluation representations, and demonstrates that all finite-dimensional irreducibles arise as subquotients of tensor products of evaluation representations, with detailed treatments of low-rank cases. The results extend the Drinfeld–Jimbo/RTT correspondence to the super and affine setting, enabling systematic classification and construction of finite-dimensional modules across all parity sequences. The approach unifies Kac-module constructions, Zhang-type standard results, and odd-reflection techniques within the RTT/affine RTT formalism, and it lays groundwork for further study of quantum affine superalgebras and their representations. The work has potential implications for supersymmetric quantum integrable systems and the broader representation theory of quantum supergroups.

Abstract

In this paper, we investigate finite-dimensional irreducible representations of the quantum affine general linear superalgebra $\mathrm{U}_q\big(\widehat{\mathfrak{gl}}_{m|n,\mathbf{s}}\big)$ for arbitrary 01-sequences $\mathbf{s}$, using the RTT presentation. We systematically construct the RTT presentation for quantum general linear superalgebra $\mathrm{U}_q\big(\mathfrak{gl}_{m|n,\mathbf{s}}\big)$, and derive a PBW basis induced by the action of the braid group, compatible with non-standard parities. We determine the necessary and sufficient conditions for the finite-dimensionality of irreducible representations of $\mathrm{U}_q\big(\mathfrak{gl}_{m|n,\mathbf{s}}\big)$ and extend the results to the affine case via the evaluation homomorphism. Specific cases such as $(m,n)=(1,1)$ demonstrate that all finite-dimensional representations are tensor products of typical evaluation representations. This work extends existing representation frameworks and classification methods to encompass arbitrary 01-sequences, establishing the foundation for subsequent research on representations of quantum affine superalgebras.

Representations of Quantum Affine General Linear Superalgebras at Arbitrary 01-Sequences

TL;DR

This work develops a comprehensive RTT-based framework for quantum affine general linear superalgebras with arbitrary 01-sequences, establishing an explicit PBW basis and transferring finite-dimensional representation theory from the standard to nonstandard parities via odd reflections. It provides necessary and sufficient conditions for the finite-dimensionality of highest-weight modules, introduces evaluation representations, and demonstrates that all finite-dimensional irreducibles arise as subquotients of tensor products of evaluation representations, with detailed treatments of low-rank cases. The results extend the Drinfeld–Jimbo/RTT correspondence to the super and affine setting, enabling systematic classification and construction of finite-dimensional modules across all parity sequences. The approach unifies Kac-module constructions, Zhang-type standard results, and odd-reflection techniques within the RTT/affine RTT formalism, and it lays groundwork for further study of quantum affine superalgebras and their representations. The work has potential implications for supersymmetric quantum integrable systems and the broader representation theory of quantum supergroups.

Abstract

In this paper, we investigate finite-dimensional irreducible representations of the quantum affine general linear superalgebra for arbitrary 01-sequences , using the RTT presentation. We systematically construct the RTT presentation for quantum general linear superalgebra , and derive a PBW basis induced by the action of the braid group, compatible with non-standard parities. We determine the necessary and sufficient conditions for the finite-dimensionality of irreducible representations of and extend the results to the affine case via the evaluation homomorphism. Specific cases such as demonstrate that all finite-dimensional representations are tensor products of typical evaluation representations. This work extends existing representation frameworks and classification methods to encompass arbitrary 01-sequences, establishing the foundation for subsequent research on representations of quantum affine superalgebras.

Paper Structure

This paper contains 24 sections, 27 theorems, 173 equations, 1 table.

Table of Contents

  1. Introduction
  2. Notations and sets up
  3. Irreducible representations of quantum general linear superalgebra
  4. Two equivalent presentations of quantum general linear superalgebra
  5. Braid group actions on $\mathrm{U}_q(\mathfrak{g}_{\mathbf{s}})$
  6. PBW basis of $\mathrm{U}_q(\mathfrak{g}_{\mathbf{s}})$
  7. Finite-dimensional irreducible representations of $\mathrm{U}_q(\mathfrak{g}_{\mathbf{s}})$
  8. Kac module over $\mathrm{U}_q(\mathfrak{g}_{\mathbf{s}})$
  9. Zhang's results for $\mathrm{U}_q(\mathfrak{g}_{\mathbf{s}})$ at $\mathbf{s}=\mathbf{s}^{\rm st}$
  10. Transition rules via odd reflections
  11. Typical and atypical irreducible representations of $\mathrm{U}_q(\mathfrak{g}_{\mathbf{s}})$
  12. Nonstandard Young-like diagram
  13. Quantum affine general linear superalgebras
  14. RTT presentation $\mathrm{U}_q(\widehat{\mathfrak{g}}_{\mathbf{s}})$
  15. PBW basis of $\mathrm{U}_q(\widehat{\mathfrak{g}}_{\mathbf{s}})$
  16. ...and 9 more sections

Key Result

Proposition 3.5

The assignment extends to a Hopf superalgebra isomorphism $\iota_{\mathbf{s}}: \mathrm{U}q(\mathfrak{g}_{\mathbf{s}}) \to \mathcal{U}q(\mathfrak{g}_{\mathbf{s}})$.

Theorems & Definitions (57)

  • Definition 3.1
  • Remark 3.2
  • Definition 3.3
  • Remark 3.4
  • Proposition 3.5
  • Proposition 3.6
  • Proposition 3.7
  • proof
  • Theorem 3.8
  • Theorem 3.9
  • ...and 47 more