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

Journal article, 2023

Lie subalgebras of L=g((x))×g[x]/xng[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.