Inflations for representations of shifted quantum affine algebras
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866908745030696960 |
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| author | Pinet, Théo |
| author_facet | Pinet, Théo |
| contents | Fix a finite-dimensional simple Lie algebra $\mathfrak{g}$ and let $\mathfrak{g}_J\subseteq\mathfrak{g}$ be a Lie subalgebra coming from a Dynkin diagram inclusion. Then, the corresponding restriction functor is not essentially surjective on finite-dimensional simple $\mathfrak{g}_J$-modules. In this article, we study Finkelberg-Tsymbaliuk's shifted quantum affine algebras $U_q^μ(\mathfrak{g})$ and the associated categories $\mathcal{O}^μ$ (defined by Hernandez). In particular, we introduce natural subalgebras $U_q^ν(\mathfrak{g}_J)\,{\subseteq}\,U_q^μ(\mathfrak{g})$ and obtain a functor $\mathcal{R}_J$ from $\mathcal{O}^{sh}\,{=}\bigoplus_μ\mathcal{O}^μ$ to $\bigoplus_ν(U_q^ν(\mathfrak{g}_J)\text{-Mod})$ using the canonical restriction functors. We then establish that $\mathcal{R}_J$ is essentially surjective on finite-dimensional simple objects by constructing notable preimages that we call inflations. We conjecture that all simple objects in $\mathcal{O}^{sh}_J$ (which is the analog of $\mathcal{O}^{sh}$ for the subalgebras $U_q^ν(\mathfrak{g}_J)$) admit some inflation and prove this for $\mathfrak{g}$ of type A-B or $\mathfrak{g}_J$ a direct sum of copies of $\mathfrak{sl}_2$ and $\mathfrak{sl}_3$. We finally apply our results to deduce certain $R$-matrices and examples of cluster structures over Grothendieck rings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_02253 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Inflations for representations of shifted quantum affine algebras Pinet, Théo Representation Theory Quantum Algebra 13F60, 16D90, 16T20, 16T25, 17B80 Fix a finite-dimensional simple Lie algebra $\mathfrak{g}$ and let $\mathfrak{g}_J\subseteq\mathfrak{g}$ be a Lie subalgebra coming from a Dynkin diagram inclusion. Then, the corresponding restriction functor is not essentially surjective on finite-dimensional simple $\mathfrak{g}_J$-modules. In this article, we study Finkelberg-Tsymbaliuk's shifted quantum affine algebras $U_q^μ(\mathfrak{g})$ and the associated categories $\mathcal{O}^μ$ (defined by Hernandez). In particular, we introduce natural subalgebras $U_q^ν(\mathfrak{g}_J)\,{\subseteq}\,U_q^μ(\mathfrak{g})$ and obtain a functor $\mathcal{R}_J$ from $\mathcal{O}^{sh}\,{=}\bigoplus_μ\mathcal{O}^μ$ to $\bigoplus_ν(U_q^ν(\mathfrak{g}_J)\text{-Mod})$ using the canonical restriction functors. We then establish that $\mathcal{R}_J$ is essentially surjective on finite-dimensional simple objects by constructing notable preimages that we call inflations. We conjecture that all simple objects in $\mathcal{O}^{sh}_J$ (which is the analog of $\mathcal{O}^{sh}$ for the subalgebras $U_q^ν(\mathfrak{g}_J)$) admit some inflation and prove this for $\mathfrak{g}$ of type A-B or $\mathfrak{g}_J$ a direct sum of copies of $\mathfrak{sl}_2$ and $\mathfrak{sl}_3$. We finally apply our results to deduce certain $R$-matrices and examples of cluster structures over Grothendieck rings. |
| title | Inflations for representations of shifted quantum affine algebras |
| topic | Representation Theory Quantum Algebra 13F60, 16D90, 16T20, 16T25, 17B80 |
| url | https://arxiv.org/abs/2404.02253 |