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On higher spin $ U_q(\operatorname{sl}_2)$-invariant $ R$-matrices


Author: A. G. Bytsko
Translated by: the author
Original publication: Algebra i Analiz, tom 17 (2005), nomer 3.
Journal: St. Petersburg Math. J. 17 (2006), 393-408
MSC (2000): Primary 81R50
DOI: https://doi.org/10.1090/S1061-0022-06-00910-1
Published electronically: March 9, 2006
MathSciNet review: 2167842
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Abstract: The spectral decomposition of regular $ U_q(\operatorname{sl}_2)$-invariant solutions of the Yang-Baxter equation is studied. An algorithm for the search of all possible spin $ s$ solutions is developed, also allowing reconstruction of the $ R$-matrix by a given nearest neighbor spin chain Hamiltonian. The algorithm is based on reduction of the Yang-Baxter equation to certain subspaces. As an application, a complete list of nonequivalent regular $ U_q(\operatorname{sl}_2)$-invariant $ R$-matrices is obtained for generic $ q$ and spins $ s\leq\frac{3}{2}$. Some further results about spectral decompositions for higher spins are also proved. In particular, it is shown that certain types of regular $ \operatorname{sl}_2$-invariant $ R$-matrices have no $ U_q(\operatorname{sl}_2)$-invariant counterparts.


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Additional Information

A. G. Bytsko
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russia
Email: bytsko@pdmi.ras.ru

DOI: https://doi.org/10.1090/S1061-0022-06-00910-1
Keywords: Quantum Lie algebra, Hopf algebra, spin, Yang--Baxter equation
Published electronically: March 9, 2006
Additional Notes: Supported by the INTAS (grant YS-03-55-962) and by RFBR (grant nos. 02-01-00085 and 03-01-00593)
Dedicated: Dedicated to Professor L. D. Faddeev on the occasion of his 70th birthday
Article copyright: © Copyright 2006 American Mathematical Society

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