Differential geometry on SU(N): Left and right invariant vector fields and one-forms

TL;DR

This paper presents an analytical method for explicitly computing invariant vector fields and one-forms on SU(N) groups using coset parametrization, facilitating the calculation of Haar measures with an example on SU(2).

Contribution

It introduces a systematic approach for calculating invariant structures on SU(N), enhancing analytical tools for group theory applications.

Findings

Explicit formulas for invariant vector fields and one-forms on SU(N)

Calculation of Haar measure using derived invariant structures

Illustrative example on SU(2) demonstrating the method

Abstract

In this paper we provide an analytical procedure for explicit calculation of the left and right invariant vector fields and one-forms on SU(N) manifold. The calculations are based on the coset parametrization of SU(N) group. The results enable us to calculate the invariant measure or Haar measure on the group. As an illustrative example, we calculate invariant vector fields and one-forms on SU(2) group.