Transfer Matrices of Rational Spin Chains via Novel BGG-Type Resolutions
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Author(s)
Frassek, Rouven
Karpov, Ivan
Tsymbaliuk, Alexander
Date Issued
February 10, 2023
Publisher
Springer Berlin Heidelberg
Citation
Frassek, Rouven, Karpov, Ivan and Tsymbaliuk, Alexander. 2023. "Transfer Matrices of Rational Spin Chains via Novel BGG-Type Resolutions."
Version
Author's final manuscript
Abstract
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1007/s00220-022-04620-6
