Toolbox of Para-differential Calculus on Compact Lie Groups

TL;DR

This paper develops a comprehensive para-differential calculus toolbox for compact Lie groups, utilizing representation theory to facilitate symbolic calculus and analyze non-local, nonlinear operators on symmetric manifolds.

Contribution

It introduces a global, coordinate-free framework for para-differential operators on compact Lie groups, with explicit formulas based on representation theory.

Findings

Provides exact symbolic calculus formulas for compact Lie groups

Constructs para-differential operators in a global, coordinate-free manner

Enhances understanding of non-local, nonlinear differential operators on symmetric manifolds

Abstract

This paper provides a toolbox of para-differential calculus on compact Lie groups. The toolbox is based on representation theory of compact Lie groups and contains exact formulas of symbolic calculus. Para-differential operators are constructed in a global, coordinate-free manner, giving lower order terms in symbolic calculus a clear form. The toolbox helps to understand non-local, nonlinear differential operators defined on certain manifolds with high symmetry.