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Symmetric algebras and Yang-Baxter equation. (English) Zbl 0884.17007

Slovák, Jan (ed.), Proceedings of the 16th Winter School on geometry and physics, Srní, Czech Republic, January 13–20, 1996. Palermo: Circolo Matematico di Palermo, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 46, 15-28 (1997).
Let \(U\) be an open subset of the complex plane, and let \(L\) denote a finite-dimensional complex simple Lie algebra. A. A. Belavin and V. G. Drinfel’d investigated non-degenerate meromorphic functions from \(U\times U\) into \(L\otimes L\) which are solutions of the classical Yang-Baxter equation [Funct. Anal. Appl. 16, 159-180 (1983; Zbl 0504.22016)]. They found that (up to equivalence) the solutions depend only on the difference of the two variables and that their set of poles forms a discrete (additive) subgroup \(\Gamma\) of the complex numbers (of rank at most 2). If \(\Gamma\) is non-trivial, they were able to completely classify all possible solutions. If \(\Gamma\) is trivial, the solutions are called rational and for \(L= sl_n(\mathbb{C})\) they were classified by A. Stolin [in Math. Scand. 69, No. 1, 57-80 (1991; Zbl 0727.17005)].
A Lie algebra \(L\) is called symmetric if there exists a non-degenerate symmetric invariant bilinear form on \(L\). In the paper under review the authors consider rational solutions of the classical Yang-Baxter equation for symmetric Lie algebras and apply this to \(L= sl_n (\mathbb{C} [\varepsilon]/ (\varepsilon^2))\) and \(L= gl_n(\mathbb{C})\). In a final section they construct rational solutions of the quantum Yang-Baxter equation for symmetric associative algebras using V. G. Drinfel’d’s quantization of a constant solution of the classical Yang-Baxter equation [Sov. Math., Dokl. 28, 667-671 (1983; Zbl 0553.58038)] and a result of the authors’ in [On Frobenius algebras and the quantum Yang-Baxter equation, Trans. Am. Math. Soc. 349, 3823-3836 (1997)].
For the entire collection see [Zbl 0866.00050].

MSC:

17B37 Quantum groups (quantized enveloping algebras) and related deformations
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
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