Topological Manin pairs and (n,s)-type series.

Lie subalgebras of L = g ((x)) × g [ x ] / x n g [ x ] , complementary to the diagonal embedding Δ of g [ [ x ] ] and Lagrangian with respect to some particular form, are in bijection with formal classical r-matrices and topological Lie bialgebra structures on the Lie algebra of formal power series g [ [ x ] ] . In this work we consider arbitrary subspaces of L complementary to Δ and associate them with so-called series of type (n, s). We prove that Lagrangian subspaces are in bijection with skew-symmetric (n, s) -type series and topological quasi-Lie bialgebra structures on g [ [ x ] ] . Using the classificaiton of Manin pairs we classify up to twisting and coordinate transformations all quasi-Lie bialgebra structures. Series of type (n, s) , solving the generalized classical Yang-Baxter equation, correspond to subalgebras of L. We discuss their possible utility in the theory of integrable systems. [ABSTRACT FROM AUTHOR]