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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2211.09779 |
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| _version_ | 1866918019455778816 |
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| author | Frenkel, Edward Hernandez, David |
| author_facet | Frenkel, Edward Hernandez, David |
| contents | We define an action of the Weyl group W of a simple Lie algebra g on a completion of the ring Y, which is the codomain of the q-character homomorphism of the corresponding quantum affine algebra U_q(g^). We prove that the subring of W-invariants of Y is precisely the ring of q-characters, which is isomorphic to the Grothendieck ring of the category of finite-dimensional representations of U_q(g^). This resolves an old puzzle in the theory of q-characters. We also identify the screening operators, which were previously used to describe the ring of q-characters, as the subleading terms of simple reflections from W in a certain limit. Our results have already found applications to the study of the category O of representations of the Borel subalgebra of U_q(g^) in arXiv:2312.13256 and to the categorification of cluster algebras in arXiv:2401.04616. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_09779 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Weyl group symmetry of q-characters Frenkel, Edward Hernandez, David Quantum Algebra Statistical Mechanics High Energy Physics - Theory Representation Theory Exactly Solvable and Integrable Systems We define an action of the Weyl group W of a simple Lie algebra g on a completion of the ring Y, which is the codomain of the q-character homomorphism of the corresponding quantum affine algebra U_q(g^). We prove that the subring of W-invariants of Y is precisely the ring of q-characters, which is isomorphic to the Grothendieck ring of the category of finite-dimensional representations of U_q(g^). This resolves an old puzzle in the theory of q-characters. We also identify the screening operators, which were previously used to describe the ring of q-characters, as the subleading terms of simple reflections from W in a certain limit. Our results have already found applications to the study of the category O of representations of the Borel subalgebra of U_q(g^) in arXiv:2312.13256 and to the categorification of cluster algebras in arXiv:2401.04616. |
| title | Weyl group symmetry of q-characters |
| topic | Quantum Algebra Statistical Mechanics High Energy Physics - Theory Representation Theory Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/2211.09779 |