QUANTUM
   INTEGRABLE
   SYSTEMS and
   GEOMETRY
   2012









 3-7 SEPTEMBER 2012
 OLHÃO, PORTUGAL

Quadratic Poisson structures on free associative algebra representations and double brackets

Vladimir Rubtsov

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Quadratic Poisson brackets on the free associative algebra representations and their pencils with a linear Poisson structure are studied. This structures are related to M. Van den Bergh double brackets and satisfied some associative analogues of Yang-Baxter equation (AYBE). We describe some classification results and new examples of constant and rational double brackets arising from solutions of various AYBE's (A. Polishchuk - T. Schedler and A. Odesskii - V. Sokolov).