Left-covariant differential calculi on ${\displaystyle S{L}_q\left(N\right)}$

We study ${\displaystyle N^2-1}$ dimensional left-covariant differential calculi on the quantum group ${\displaystyle S{L}_q\left(N\right)}$. 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 ${\displaystyle N^2-1ch\infty sek}$ as in the case of the corresponding classical Lie group SL(N).