Parabolic quantum affine algebras
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910957889912832 |
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| author | Bostanci, Kudret Kus, Deniz |
| author_facet | Bostanci, Kudret Kus, Deniz |
| contents | Maximal parabolic subalgebras of untwisted affine Kac-Moody algebras were studied in the context of Borel-de Siebenthal theory in [13], where they were realized as certain equivariant map algebras with a non-free abelian group action. In this paper, we show that this perspective naturally extends to non-maximal parabolic subalgebras and introduce their quantum analogues - called parabolic quantum affine algebras - in analogy with ordinary quantum affine algebras and their classical counterpart, the loop algebra. While the definition in the Drinfeld-Jimbo presentation is straightforward, the realization in Drinfeld's second presentation requires quantum root vectors associated not only to simple roots but also to certain non-simple roots. A distinguished positive root $γ_0$ plays a central role in all constructions. Along the way, we construct a PBW-type basis, establish a second triangular decomposition, and determine the action of the braid group on the Cartan part of the algebra via Lusztig's automorphisms. Finally, we classify the finite-dimensional irreducible representations under a technical condition on $γ_0$, referred to as repetition-free, in terms of Drinfeld polynomials with some additional data. The key difference from the ordinary quantum affine case is that the degrees of the polynomials are only bounded by a certain highest weight, rather than being uniquely determined by it. In the maximal parabolic case, the classification can alternatively be phrased in terms of Drinfeld polynomials satisfying certain divisibility conditions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_14624 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Parabolic quantum affine algebras Bostanci, Kudret Kus, Deniz Quantum Algebra Representation Theory Maximal parabolic subalgebras of untwisted affine Kac-Moody algebras were studied in the context of Borel-de Siebenthal theory in [13], where they were realized as certain equivariant map algebras with a non-free abelian group action. In this paper, we show that this perspective naturally extends to non-maximal parabolic subalgebras and introduce their quantum analogues - called parabolic quantum affine algebras - in analogy with ordinary quantum affine algebras and their classical counterpart, the loop algebra. While the definition in the Drinfeld-Jimbo presentation is straightforward, the realization in Drinfeld's second presentation requires quantum root vectors associated not only to simple roots but also to certain non-simple roots. A distinguished positive root $γ_0$ plays a central role in all constructions. Along the way, we construct a PBW-type basis, establish a second triangular decomposition, and determine the action of the braid group on the Cartan part of the algebra via Lusztig's automorphisms. Finally, we classify the finite-dimensional irreducible representations under a technical condition on $γ_0$, referred to as repetition-free, in terms of Drinfeld polynomials with some additional data. The key difference from the ordinary quantum affine case is that the degrees of the polynomials are only bounded by a certain highest weight, rather than being uniquely determined by it. In the maximal parabolic case, the classification can alternatively be phrased in terms of Drinfeld polynomials satisfying certain divisibility conditions. |
| title | Parabolic quantum affine algebras |
| topic | Quantum Algebra Representation Theory |
| url | https://arxiv.org/abs/2505.14624 |