Computing cohomology spaces of left-invariant involutive structures on $\mathrm{SU}(2)$: examples

TL;DR

This paper investigates the cohomology spaces of left-invariant involutive structures on SU(2), providing explicit computations and conditions for closedness of certain operators, with applications of representation theory.

Contribution

It offers explicit calculations of cohomology spaces and closedness conditions for involutive structures on SU(2), illustrating applications of prior theoretical results.

Findings

Determined closedness of the range of a complex vector field on SU(2)

Computed smooth cohomology spaces for a corank 1 structure

Provided concrete examples illustrating the application of representation theory

Abstract

In these notes we study left-invariant involutive structures on $\mathrm{SU}(2)$, the most na\"ive non-commutative compact Lie group. We determine closedness of the range (in the smooth topology) of a single complex vector field spanning the standard CR structure of $\mathrm{SU}(2)$ and also compute the smooth cohomology spaces of a corank $1$ structure. In our approach, it is fundamental to understand concretely the irreducible representations of the ambient Lie group and how left-invariant vector fields operate on their matrix coefficients (which we borrow from the book of Ruzhansky and Turunen (2010)). Our purpose is solely to provide some easy applications of the theory developed in a previous paper (2019), as the results shown here can probably be obtained by more direct methods.