Birkhoff decompositions for loop groups with coefficient algebras

TL;DR

This paper extends Birkhoff decomposition theory to loop groups with coefficients in commutative topological algebras, including Banach-Lie groups, broadening the scope of classical loop group analysis.

Contribution

It introduces new Birkhoff decomposition results for loop groups with scalar extensions in topological algebras, including Banach-Lie groups, expanding existing frameworks.

Findings

Birkhoff decompositions established for loop groups with algebra-valued coefficients

Results applicable to complex Banach-Lie group-valued loops

Extension of classical loop group theory to algebraic coefficient settings

Abstract

Starting with a finite-dimensional complex Lie algebra, we extend scalars using suitable commutative topological algebras. We study Birkhoff decompositions for the corresponding loop groups. Some results remain valid for loop groups with valued in complex Banach-Lie groups.