Weyl group twists and representations of quantum affine Borel algebras
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866929318327746560 |
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| author | Wang, Keyu |
| author_facet | Wang, Keyu |
| contents | We define categories $\mathcal{O}^w$ of representations of Borel subalgebras $\mathcal{U}_q\mathfrak{b}$ of quantum affine algebras $\mathcal{U}_q\hat{\mathfrak{g}}$, which come from the category $\mathcal{O}$ twisted by Weyl group elements $w$. We construct inductive systems of finite-dimensional $\mathcal{U}_q\mathfrak{b}$-modules twisted by $w$, which provide representations in the category $\mathcal{O}^w$. We also establish a classification of simple modules in these categories $\mathcal{O}^w$.
We explore convergent phenomenon of $q$-characters of representations of quantum affine algebras, which conjecturally give the $q$-characters of representations in $\mathcal{O}^w$.
Furthermore, we propose a conjecture concerning the relationship between the category $\mathcal{O}$ and the twisted category $\mathcal{O}^w$, and we propose a possible connection with shifted quantum affine algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_11749 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Weyl group twists and representations of quantum affine Borel algebras Wang, Keyu Representation Theory Quantum Algebra We define categories $\mathcal{O}^w$ of representations of Borel subalgebras $\mathcal{U}_q\mathfrak{b}$ of quantum affine algebras $\mathcal{U}_q\hat{\mathfrak{g}}$, which come from the category $\mathcal{O}$ twisted by Weyl group elements $w$. We construct inductive systems of finite-dimensional $\mathcal{U}_q\mathfrak{b}$-modules twisted by $w$, which provide representations in the category $\mathcal{O}^w$. We also establish a classification of simple modules in these categories $\mathcal{O}^w$. We explore convergent phenomenon of $q$-characters of representations of quantum affine algebras, which conjecturally give the $q$-characters of representations in $\mathcal{O}^w$. Furthermore, we propose a conjecture concerning the relationship between the category $\mathcal{O}$ and the twisted category $\mathcal{O}^w$, and we propose a possible connection with shifted quantum affine algebras. |
| title | Weyl group twists and representations of quantum affine Borel algebras |
| topic | Representation Theory Quantum Algebra |
| url | https://arxiv.org/abs/2404.11749 |