On higher spin 𝑈_{𝑞}(𝑠𝑙₂)-invariant 𝑅-matrices

Bytsko, A.

doi:10.1090/S1061-0022-06-00910-1

On higher spin $U_q(\operatorname {sl}_2)$-invariant $R$-matrices
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by A. G. Bytsko
Translated by: the author

St. Petersburg Math. J. 17 (2006), 393-408

DOI: https://doi.org/10.1090/S1061-0022-06-00910-1

Published electronically: March 9, 2006

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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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