Classification of Left-Covariant Differential Calculi on the Quantum Group SLq(2),

For transcendental values of q the quantum tangent spaces of all left-covariant first order differential calculi of dimension less than four on the quantum group SLq (2) are given. All such differential calculi Γ are determined and investigated for which the left-invariant differential one-forms ω(u12), ω(u21), and ω(u11 − u22) generate Γ as a bimodule and the universal higher order differential calculus has the same dimension as in the classical case. Important properties (cohomology spaces, *-structures, braidings, generalized Lie brackets) of these calculi are examined as well.