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

Lie subalgebras of $L=\mathfrak{g}(\text{\hspace{0.17em}⁣}(x)\text{\hspace{0.17em}⁣})\times \mathfrak{g}[x]\mathrm{/}{x}^n\mathfrak{g}[x]$, complementary to the diagonal embedding $\mathrm{\Delta }$ of $\mathfrak{g}[\text{\hspace{0.17em}⁣}[x]\text{\hspace{0.17em}⁣}]$ 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 $\mathfrak{g}[\text{\hspace{0.17em}⁣}[x]\text{\hspace{0.17em}⁣}]$. In this work we consider arbitrary subspaces of $L$ complementary to $\mathrm{\Delta }$ 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 $\mathfrak{g}[\text{\hspace{0.17em}⁣}[x]\text{\hspace{0.17em}⁣}]$. 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 Yang-Baxter equation, correspond to subalgebras of $L$. We discuss their possible utility in the theory of integrable systems.