Representations of Quantum Affine General Linear Superalgebras at Arbitrary 01-Sequences
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914134284566528 |
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| author | Lin, Hongda Zhang, Honglian |
| author_facet | Lin, Hongda Zhang, Honglian |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_02393 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Representations of Quantum Affine General Linear Superalgebras at Arbitrary 01-Sequences Lin, Hongda Zhang, Honglian Quantum Algebra Representation Theory 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. |
| title | Representations of Quantum Affine General Linear Superalgebras at Arbitrary 01-Sequences |
| topic | Quantum Algebra Representation Theory |
| url | https://arxiv.org/abs/2511.02393 |