Left-Covariant Differential Calculi on SL_q(2) and SL_q(3)

Abstract

We study $N^2-1$ dimensional left-covariant differential calculi on the quantum group $SL_q(N)$ for which the generators of the quantum Lie algebras annihilate the quantum trace. In this way we obtain one distinguished calculus on $SL_q(2)$ (which corresponds to Woronowicz' 3D-calculus on $SU_q(2)$) and two distinguished calculi on $SL_q(3)$ such that the higher order calculi give the ordinary differential calculus on $SL(2)$ and $SL(3)$, respectively, in the limit $q\rightarrow 1$. Two new differential calculi on $SL_q(3)$ are introduced and developed in detail.