New \(r\)-matrices for Lie bialgebra structures over polynomials (Q993443)

Let \(\mathfrak g\) be a finite-dimensional simple complex Lie algebra. The Lie bialgebra structures on \(\mathfrak g[u]\) were discussed by \textit{F. Montaner, A. Stolin} and \textit{E. Zelmanov} [Sel. Math., New Ser. 16, No. 4, 935--962 (2010; Zbl 1210.17024), see also \url{arXiv:1001.4824}]. Every Lie bialgebra structure on \(\mathfrak g[u]\) corresponds to a unique bounded Lagrangian subalgebra \(W\) of one of (1) \(\mathfrak g((u))\), (2) \(\mathfrak g((u))\oplus \mathfrak g\) or (3) \(\mathfrak g((u))\oplus (\mathfrak g+\varepsilon \mathfrak g)\) with \(\varepsilon^2=0\). The paper under review discusses these structures and corresponding \(r\)-matrices in the case where \(W\) is contained in the maximal order corresponding to the negative of a maximal root (for other arbitrary simple roots, the authors promise to treat these in a subsequent article). The details cannot be conveniently listed here, but we do note that in all cases, the \(r\)-matrix satisfies the classical Yang-Baxter equation. The authors introduce the notion of a quasi-twist equivalence between two Lie bialgebra structures on \(\mathfrak g[u]\), and prove that certain Lie bialgebra structures are quasi-twist equivalent.