Twisted $\text{SU}(2)$ Group. An Example of a Non-Commutative Differential Calculus

Abstract

For any number $\nu$ in the interval $[-1,1]$, a $C^{\ast }$-algebra $A$, generated by two elements $\alpha$ and $\gamma$ satisfying simple (depending on $\nu$) commutation relation, is introduced and investigated.

If $\nu =1$, then the algebra coincides with the algebra of all continuous functions on the group $\text{SU}(2)$. Therefore, one can introduce many notions related to the fact that $\text{SU}(2)$ is a Lie group. In particular one can speak about convolution products, Haar measure, differential structure, cotangent boundle, left invariant differential forms. Lie brackets, infinitesimal shifts and Cartan Maurer formulae. One can also consider representations of $\text{SU}(2)$.

For $\nu <1$, the algebra $A$ is no longer commutative, however the notions listed above are meaningful. Therefore, $A$ can be considered as the algebra of all “continuous functions” on a “pseudospace ${\text{S}}_{\nu }\text{U}(2)$” and this pseudospace is endowed with a Lie group structure.

The potential applications to the quantum physics are indicated.