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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.

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