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 $\omega (u_2^1)$, $\omega (u_1^2)$ 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

LaTeX2e, 43 pages, to appear in Journal of Algebra