Classification of Left-Covariant Differential Calculi on the Quantum Group $\SLq 2$

Abstract

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 $\Gamma $ are determined and investigated for which the left-invariant differential one-forms $\omega (u^1_2)$, $\omega (u^2_1)$ and $\omega (u^1_1-u^2_2)$ generate $\Gamma $ 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. Keywords: quantum groups, noncommutative differential calculus, quantum tangent space