Classification of left-covariant differential calculi on the quantum groupSLq(2)

For transcendental values of q the quantum tangent spaces of all left-covar iant first order differential calculi of dimension less than four on the qu antum group SLq(2) are given. All such differential calculi Gamma are deter mined and investigated for which the left-invariant differential one-forms omega (u(2)(1)), omega (u(1)(2)), and omega (u(1)(1) - u(2)(2)) generate Ga mma as a bimodule and the universal higher order differential calculus has the same dimension as in the classical case. Important properties (cohomolo gy spaces, *-structures, braidings, generalized Lie brackets) of these calc uli are examined as well, (C) 2001 Academic Press.