Smooth Structures, Actions of the Lie Algebra $\mathfrak{su}(2)$ and Haar Measures on Non-Commutative Three Dimensional Spheres

Abstract

We study a smooth structure on a non-commutative 3-sphere $S_\Theta^3$ defined as a deformed $C^*$-algebra of $C(S^3)$ by a continuous function $\Theta$. We then consider the subalgebra $(S_\Theta^3)^\infty$ of all smooth elements of $S_\Theta^3$. It is a non-commutative version of $S^{3}$ as a smooth manifold. We also construct a smooth linear map from $(S_\Theta^3)^\infty$ to the algebra $C^\infty(S^3)$ of all smooth functions on $S^3$ so that the Lie algebra $\mathfrak{su}(2)$ acts on $(S_\Theta^3)^\infty$ with a twisted Leibniz's rule. Finally we find a Haar measure on $S_\Theta^3$ and show its uniqueness.