Compatible Lie Brackets and the Yang-Baxter Equation |
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Authors: | I Z Golubchik V V Sokolov |
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Institution: | (1) Bashkir State Pedagogical University, Ufa, Russia;(2) Landau Institute for Theoretical Physics, RAS, Moscow, Russia |
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Abstract: | We show that any pair of compatible Lie brackets with a common invariant form produces a nonconstant solution of the classical
Yang-Baxter equation. We describe the corresponding Poisson brackets, Manin triples, and Lie bialgebras. It turns out that
all bialgebras associated with the solutions found by Belavin and Drinfeld are isomorphic to some bialgebras generated by
our solutions. For any compatible pair, we construct a double with a common invariant form and find the corresponding solution
of the quantum Yang-Baxter equation for this double.
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Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 2, pp. 195–207, February, 2006. |
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Keywords: | Yang-Baxter equation Lie bialgebra Manin triple |
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