Bicovariant differential calculi on \(S_ \mu U(2)\) (Q1196097)

The author investigates bicovariant differential calculi on the quantum group (pseudogroup) \(SU_ \mu(2)\) in accordance with the general theory due to \textit{S. L. Woronowicz} [Commun. Math. Phys. 122, 125-170 (1989; Zbl 0751.58042)]. It is shown that there exists no three-dimensional calculus and that there exist exactly two four-dimensional calculi \(4D_ \pm\). The two calculi are introduced by describing right ideals \(R_ \pm\) in \(\ker(e)\) (\(e\) is the counit). Afterwards all steps of the theory are applied to the calculi \(4D_ \pm\) and elaborated quite explicitly and in full detail. This includes specification of a basis in \(_{\text{inv}}\Gamma\) (left invariant forms) and calculations of external products, external derivatives and also the commutators \([\chi_ j,\chi_ k]\) where \(\{\chi_ j\}_{- 1}^ 4\) is a basis in the dual to \(_{\text{inv}}\Gamma\). The classical limit is obtained as well and the resulting differential operators \(\chi_ j\) on \(SU(2)\) differ for the cases \(4D_ +\) and \(4D_ -\).