Transfer Matrices of Rational Spin Chains via Novel BGG-Type Resolutions
Abstract We obtain BGG-type formulas for transfer matrices of irreducible finite-dimensional representations of the classical Lie algebras $${\mathfrak {g}}$$ g...
| Main Authors: | , , |
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| Other Authors: | |
| Format: | Article |
| Language: | English |
| Published: |
Springer Berlin Heidelberg
2023
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| Online Access: | https://hdl.handle.net/1721.1/150612 |
| _version_ | 1811077747343097856 |
|---|---|
| author | Frassek, Rouven Karpov, Ivan Tsymbaliuk, Alexander |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Frassek, Rouven Karpov, Ivan Tsymbaliuk, Alexander |
| author_sort | Frassek, Rouven |
| collection | MIT |
| description | Abstract
We obtain BGG-type formulas for transfer matrices of irreducible finite-dimensional representations of the classical Lie algebras
$${\mathfrak {g}}$$
g
, whose highest weight is a multiple of a fundamental one and which can be lifted to the representations over the Yangian
$$Y({\mathfrak {g}})$$
Y
(
g
)
. These transfer matrices are expressed in terms of transfer matrices of certain infinite-dimensional highest weight representations (such as parabolic Verma modules and their generalizations) in the auxiliary space. We further factorise the corresponding infinite-dimensional transfer matrices into the products of two Baxter Q-operators, arising from our previous study Frassek et al. (Adv. Math. 401:108283, 2022), Frassek and Tsymbaliuk (Commun. Math. Phys. 392:545–619, 2022) of the degenerate Lax matrices. Our approach is crucially based on the new BGG-type resolutions of the finite-dimensional
$${\mathfrak {g}}$$
g
-modules, which naturally arise geometrically as the restricted duals of the Cousin complexes of relative local cohomology groups of ample line bundles on the partial flag variety G/P stratified by
$$B_{-}$$
B
-
-orbits. |
| first_indexed | 2024-09-23T10:47:56Z |
| format | Article |
| id | mit-1721.1/150612 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T10:47:56Z |
| publishDate | 2023 |
| publisher | Springer Berlin Heidelberg |
| record_format | dspace |
| spelling | mit-1721.1/1506122024-02-20T05:59:55Z Transfer Matrices of Rational Spin Chains via Novel BGG-Type Resolutions Frassek, Rouven Karpov, Ivan Tsymbaliuk, Alexander Massachusetts Institute of Technology. Department of Mathematics Abstract We obtain BGG-type formulas for transfer matrices of irreducible finite-dimensional representations of the classical Lie algebras $${\mathfrak {g}}$$ g , whose highest weight is a multiple of a fundamental one and which can be lifted to the representations over the Yangian $$Y({\mathfrak {g}})$$ Y ( g ) . These transfer matrices are expressed in terms of transfer matrices of certain infinite-dimensional highest weight representations (such as parabolic Verma modules and their generalizations) in the auxiliary space. We further factorise the corresponding infinite-dimensional transfer matrices into the products of two Baxter Q-operators, arising from our previous study Frassek et al. (Adv. Math. 401:108283, 2022), Frassek and Tsymbaliuk (Commun. Math. Phys. 392:545–619, 2022) of the degenerate Lax matrices. Our approach is crucially based on the new BGG-type resolutions of the finite-dimensional $${\mathfrak {g}}$$ g -modules, which naturally arise geometrically as the restricted duals of the Cousin complexes of relative local cohomology groups of ample line bundles on the partial flag variety G/P stratified by $$B_{-}$$ B - -orbits. 2023-05-08T18:00:20Z 2023-05-08T18:00:20Z 2023-02-10 2023-04-30T03:13:06Z Article http://purl.org/eprint/type/JournalArticle https://hdl.handle.net/1721.1/150612 Frassek, Rouven, Karpov, Ivan and Tsymbaliuk, Alexander. 2023. "Transfer Matrices of Rational Spin Chains via Novel BGG-Type Resolutions." en https://doi.org/10.1007/s00220-022-04620-6 Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature application/pdf Springer Berlin Heidelberg Springer Berlin Heidelberg |
| spellingShingle | Frassek, Rouven Karpov, Ivan Tsymbaliuk, Alexander Transfer Matrices of Rational Spin Chains via Novel BGG-Type Resolutions |
| title | Transfer Matrices of Rational Spin Chains via Novel BGG-Type Resolutions |
| title_full | Transfer Matrices of Rational Spin Chains via Novel BGG-Type Resolutions |
| title_fullStr | Transfer Matrices of Rational Spin Chains via Novel BGG-Type Resolutions |
| title_full_unstemmed | Transfer Matrices of Rational Spin Chains via Novel BGG-Type Resolutions |
| title_short | Transfer Matrices of Rational Spin Chains via Novel BGG-Type Resolutions |
| title_sort | transfer matrices of rational spin chains via novel bgg type resolutions |
| url | https://hdl.handle.net/1721.1/150612 |
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