Compatible Poisson brackets, quadratic Poisson algebras and classical r-matrices.

Summary. We show that for a general quadratic Poisson bracket it is possible to define a lot of associated linear Poisson brackets: its linearizations in the neighbor-hood of special points. We prove that the constructed linear Poisson brackets are always compatible with the initial quadratic Poisson bracket. We apply the obtained results to the cases of the standard quadratic r-matrix bracket and to classical “twisted reflection algebra ” brackets. In the first case we obtain that there exist only one non-equivalent linearization: standard linear r-matrix bracket and recover well-known result that the standard quadratic and linear r-matrix brackets are compatible. In the second case we show that there are a lot of non-equivalent linearizations of the classical twisted reflection algebra bracket and all of them are compatible with initial quadratic bracket