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Balinsky, A.; Burman, Yu.

Quadratic Poisson brackets and the Drinfeld theory for associative algebras. (English) Zbl 0857.16034

The known connection between skew-symmetric solutions of the classical Yang-Baxter equation for the elements \(r^{ij}\) of the tensor square of some Lie algebra \(\mathfrak g\) and Poisson Lie structures on the corresponding Lie group \(G\) (Drinfeld, 1983) is studied in detail. It is shown that the Poisson brackets compatible with multiplication in an associative algebra structure \(A\) are quadratic and correspond to the differentiation of the algebra of symmetric elements of \(A\otimes A\) (\(\text{Symm}(A\otimes A)\)). By making use of the Jacobi identity it is verified that such a differentiation satisfies some version of Yang-Baxter equation. A Poisson Lie structure is obtained by considering the restriction of the brackets under consideration to the group of invertible elements of the algebra \(A\). It is also proved in the paper (Theorem 8) that any coboundary Poisson Lie structure on a simple Lie group is a Poisson covering over Poisson Lie group whose Poisson bracket is quadratic in some global coordinate system. Finally, some examples are given.
Reviewer: A.A.Bogush (Minsk)

MSC:

16W30 Hopf algebras (associative rings and algebras) (MSC2000)
17B37 Quantum groups (quantized enveloping algebras) and related deformations
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
53D50 Geometric quantization
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References:

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