Projective Banach Lie bialgebras, the projective Yang-Baxter equation and projective Poisson-Lie groups
Zhonghua Li, Shukun Wang
TL;DR
The paper advances a projective tensor product framework for Banach Lie bialgebras by introducing projective Banach Lie bialgebras and the projective Yang–Baxter equation, then shows how projective $r$-matrices generate such bialgebras. It further defines projective Banach Poisson–Lie groups and proves their differentiation yields a projective Banach Lie bialgebra, linking geometry and algebra in the projective setting. A key contribution is showing that bounded $\mathcal{O}$-operators provide an equivalent description of triangular projective $r$-matrices, offering operator-theoretic tools to study the classical Yang–Baxter equation in this context. The results unify Banach Lie bialgebras, Poisson–Lie geometry, and $\mathcal{O}$-operators within a projective tensor framework, with potential applications to infinite-dimensional integrable systems and quantum-group theory in Banach spaces.
Abstract
In this paper, we first introduce the notion of projective Banach Lie bialgebras as the projective tensor product analogue of Banach Lie bialgebras. Then we consider the completion of the classical Yang-Baxter equation and classical r-matrices, and propose the notions of the projective Yang-Baxter equation and projective r-matrices. As in the finite-dimensional case, we prove that every quasi-triangular projective r-matrix gives rise to a projective Banach Lie bialgebra. Next adapting Poisson Banach-Lie groups to the projective tensor product setting, we propose the notion of projective Banach Poisson-Lie groups and show that the differentiation of a projective Banach Poisson-Lie group has the projective Banach Lie bialgebra structure. Finally considering bounded $\mathcal{O}$-operators on Banach Lie algebras, we give an equivalent description of triangular projective r-matrices.