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Solutions for the constant quantum Yang-Baxter equation from Lie (super)algebras. (English) Zbl 1143.81307
Summary: We present a systematic procedure to obtain singular solutions of the constant quantum Yang-Baxter equation in arbitrary dimension. This approach, inspired by the Lie (super)algebra structure, is explicitly applied to the particular case of (graded) contractions of the orthogonal real algebra $$\mathfrak{so} (N+1)$$. In this way we show that “classical” contraction parameters which appear in the commutation relations of the contracted Lie algebras, become quantum deformation parameters, arising as entries of the resulting quantum $$R$$-matrices.

##### MSC:
 81R12 Groups and algebras in quantum theory and relations with integrable systems 17B99 Lie algebras and Lie superalgebras 82B23 Exactly solvable models; Bethe ansatz