Projective Banach Lie bialgebras, the projective Yang-Baxter equation and projective Poisson-Lie groups

TL;DR

This paper introduces the concept of projective Banach Lie bialgebras, extends the classical Yang-Baxter equation to a projective setting, and explores their relation to projective Poisson-Lie groups, providing a new framework in infinite-dimensional Lie theory.

Contribution

It develops the theory of projective Banach Lie bialgebras, defines the projective Yang-Baxter equation, and connects these structures to projective Banach Poisson-Lie groups, extending finite-dimensional concepts.

Findings

Every quasi-triangular projective r-matrix yields a projective Banach Lie bialgebra.

Differentiation of a projective Banach Poisson-Lie group results in a projective Banach Lie bialgebra.

Triangular projective r-matrices can be characterized via bounded $\\mathcal{O}$-operators on Banach Lie algebras.

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.