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Abstract

We study $N^{2} - 1$ dimensional left-covariant differential calculi on the quantum group $SL_q(N)$. In this way we obtain four classes of differential calculi which are algebraically much simpler as the bicovariant calculi. The algebra generated by the left-invariant vector fields has only quadratic-linear relations and posesses a Poincaré-Birkhoff-Witt basis. We use the concept of universal (higher order) differential calculus associated with a given left-covariant first order differential calculus. It turns out that the space of left-invariant k-forms has the dimension $N^{2} - 1\choose k$ as in the case of the corresponding classical Lie group SL(N).