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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

 

 

General solution of the Yang-Baxter equation with symmetry group $ \mathrm{SL}(\mathit{n},\mathbb{C})$


Authors: S. E. Derkachev and A. N. Manashov
Translated by: B. M. Bekker
Original publication: Algebra i Analiz, tom 21 (2009), nomer 4.
Journal: St. Petersburg Math. J. 21 (2010), 513-577
MSC (2010): Primary 81R12
DOI: https://doi.org/10.1090/S1061-0022-2010-01106-3
Published electronically: May 20, 2010
MathSciNet review: 2584208
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Abstract: The problem of constructing the $ \mathrm{R}$-matrix is considered in the case of an integrable spin chain with symmetry group $ \mathrm{SL}(\mathit{n},\mathbb{C})$. A fairly complete study of general $ \mathrm{R}$-matrices acting in the tensor product of two continuous series representations of $ \mathrm{SL}(n,\mathbb{C})$ is presented. On this basis, $ \mathrm{R}$-matrices are constructed that act in the tensor product of Verma modules (which are infinite-dimensional representations of the Lie algebra $ \mathrm{sl}(n)$), and also $ \mathrm{R}$-matrices acting in the tensor product of finite-dimensional representations of the Lie algebra $ \mathrm{sl}(n)$.


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Additional Information

S. E. Derkachev
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russia
Email: derkach@pdmi.ras.ru

A. N. Manashov
Affiliation: Physics Department, St. Petersburg State University, Ulyanovskaya 3, St. Petersburg 198504, Russia and Institute for Theoretical Physics, University of Regensburg, D-93040 Regensburg, Germany
Email: alexander.manashov@physik.uni-regensburg.de

DOI: https://doi.org/10.1090/S1061-0022-2010-01106-3
Keywords: R-matrix, quantum inverse problem method, algebraic Bethe ansatz, Baxter operator
Received by editor(s): November 19, 2008
Published electronically: May 20, 2010
Additional Notes: Supported by RFBR, grants 07-02-92166-CNRS_a and 09-01-93108-CNRS_a (the first and the second author), grants 08-01-00683_a and 09-01-12150-ofi_m (the first author), National project RNP 2.1.1/1575 and German Research Foundation (DFG) grant 9209282 (the second author).
Article copyright: © Copyright 2010 American Mathematical Society