Representations of shifted quantum affine algebras
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2020
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| _version_ | 1866912091516960768 |
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| author | Hernandez, David |
| author_facet | Hernandez, David |
| contents | We develop the representation theory of shifted quantum affine algebras $\mathcal{U}_q^μ(\hat{\mathfrak{g}})$ and of their truncations which appeared in the study of quantized K-theoretic Coulomb branches of 3d $N = 4$ SUSY quiver gauge theories. Our approach is based on novel techniques, which are new in the cases of shifted Yangians or ordinary quantum affine algebras as well : realization in terms of asymptotical subalgebras of the quantum affine algebra $\mathcal{U}_q(\hat{\mathfrak{g}})$, induction and restriction functors to the category $\mathcal{O}$ of representations of the Borel subalgebra $\mathcal{U}_q(\hat{\mathfrak{b}})$ of $\mathcal{U}_q(\hat{\mathfrak{g}})$, relations between truncations and Baxter polynomiality in quantum integrable models, parametrization of simple modules via Langlands dual interpolation. We first introduce the category $\mathcal{O}_μ$ of representations of $\mathcal{U}_q^μ(\hat{\mathfrak{g}})$ and we classify its simple objects. Then we establish the existence of fusion products and we get a ring structure on the sum of the Grothendieck groups $K_0(\mathcal{O}_μ)$. We classify simple finite-dimensional representations of $\mathcal{U}_q^μ(\hat{\mathfrak{g}})$ and we obtain a cluster algebra structure on the Grothendieck ring of finite-dimensional representations. We prove a truncation has only a finite number of simple representations and we introduce a related partial ordering on simple modules. Eventually, we state a conjecture on the parametrization of simple modules of a non simply-laced truncation in terms of the Langlands dual Lie algebra. We have several evidences, including a general result for simple finite-dimensional representations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2010_06996 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Representations of shifted quantum affine algebras Hernandez, David Representation Theory High Energy Physics - Theory Quantum Algebra We develop the representation theory of shifted quantum affine algebras $\mathcal{U}_q^μ(\hat{\mathfrak{g}})$ and of their truncations which appeared in the study of quantized K-theoretic Coulomb branches of 3d $N = 4$ SUSY quiver gauge theories. Our approach is based on novel techniques, which are new in the cases of shifted Yangians or ordinary quantum affine algebras as well : realization in terms of asymptotical subalgebras of the quantum affine algebra $\mathcal{U}_q(\hat{\mathfrak{g}})$, induction and restriction functors to the category $\mathcal{O}$ of representations of the Borel subalgebra $\mathcal{U}_q(\hat{\mathfrak{b}})$ of $\mathcal{U}_q(\hat{\mathfrak{g}})$, relations between truncations and Baxter polynomiality in quantum integrable models, parametrization of simple modules via Langlands dual interpolation. We first introduce the category $\mathcal{O}_μ$ of representations of $\mathcal{U}_q^μ(\hat{\mathfrak{g}})$ and we classify its simple objects. Then we establish the existence of fusion products and we get a ring structure on the sum of the Grothendieck groups $K_0(\mathcal{O}_μ)$. We classify simple finite-dimensional representations of $\mathcal{U}_q^μ(\hat{\mathfrak{g}})$ and we obtain a cluster algebra structure on the Grothendieck ring of finite-dimensional representations. We prove a truncation has only a finite number of simple representations and we introduce a related partial ordering on simple modules. Eventually, we state a conjecture on the parametrization of simple modules of a non simply-laced truncation in terms of the Langlands dual Lie algebra. We have several evidences, including a general result for simple finite-dimensional representations. |
| title | Representations of shifted quantum affine algebras |
| topic | Representation Theory High Energy Physics - Theory Quantum Algebra |
| url | https://arxiv.org/abs/2010.06996 |