We study $N^2-1$ dimensional left-covariant differential calculi on the quantum group $S{L}_q\left(N\right)$. 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...

We find the limit distributions for a spectrum of a system of n particles governed by a k-body interaction. The hamiltonian of this system is modelled by a Gaussian random matrix. We show that the limit distribution is a q-deformed Gaussian distribution with the deformation parameter q depending on the fraction k/√n. The family of q-deformed Gaussian distributions include the Gaussian distribution and the semicircular law; therefore our result is a generalization of the results of Wigner [Wig1,...

We consider two standard group representations: one acting on functions by translations and dilations, the other by translations and modulations, and we study local Toeplitz operators based on them. Local Toeplitz operators are the averages of projection-valued functions $g\mapsto P_{g,\varphi }$, where for a fixed function ϕ, $P_{g,\varphi }$ denotes the one-dimensional orthogonal projection on the function $U_g\varphi$, U is a group representation and g is an element of the group. They are defined as integrals ${ʃ}_W{P}_{g,\varphi }dg$, where W is an open, relatively...