F_q[M_n], F_q[GL_n] and F_q[SL_n] as quantized hyperalgebras

The quantized universal enveloping algebra U_q(gl(n)) has two integral forms - over Z[q,q^{-1}] - the restricted (by Lusztig) and the unrestricted (by De Concini and Procesi) one. Dually, the quantum function algebra F_q[GL(n)] has two integral forms, namely those of all elements - of F_q[GL(n)] - which take values in Z[q,q^{-1}] when paired respectively with the restricted or the unrestricted form of U_q(gl(n)). The first one is the well-known form generated over Z[q,q^{-1}] by the entries of a q-matrix and the inverse of its quantum determinant. In this paper instead we study the second integral form, say F'_q[GL(n)], i.e. that of all elements which are Z[q,q^{-1}]-valued over the unrestricted form of U_q(gl(n)). In particular we yield a presentation of it by generators and relations, and a PBW-like theorem: in short, it is an algebra of "quantum divided powers" and "quantum binomial coefficients". Moreover, we give a direct proof that F'_q[GL(n)] is a Hopf subalgebra of F_q[GL(n)], and that its specialization at q=1 is the Z-hyperalgebra over gl(n)^*, the Lie bialgebra dual to gl(n). In addition, we describe explicitly the specializations of F'_q[GL(n)] at roots of 1, and the associated quantum Frobenius (epi)morphism. The same analysis is done for F'_q[SL(n)] and (as a key step) F'_q[Mat(n)]: in fact, for the latter the strongest results are obtained. This work extends to general n>2 the results for n=2, already treated in math.QA/0411440.Comment: AMS-TeX file, 34 pages. This is the final version (after the journal's proofs correction

Path Integrals in Noncommutative Quantum Mechanics

Extension of Feynman's path integral to quantum mechanics of noncommuting spatial coordinates is considered. The corresponding formalism for noncommutative classical dynamics related to quadratic Lagrangians (Hamiltonians) is formulated. Our approach is based on the fact that a quantum-mechanical system with a noncommutative configuration space may be regarded as another effective system with commuting spatial coordinates. Since path integral for quadratic Lagrangians is exactly solvable and a general formula for probability amplitude exists, we restricted our research to this class of Lagrangians. We found general relation between quadratic Lagrangians in their commutative and noncommutative regimes. The corresponding noncommutative path integral is presented. This method is illustrated by two quantum-mechanical systems in the noncommutative plane: a particle in a constant field and a harmonic oscillator.Comment: 9 pages,some misprints correcte

New classes of weighted H\"older-Zygmund spaces and the wavelet transform

We provide a new and elementary proof of the continuity theorem for the wavelet and left-inverse wavelet transforms on the spaces $\mathcal{S}_0(\mathbb{R}^n)$ and $\mathcal{S}(\mathbb{H}^{n+1})$. We then introduce and study a new class of weighted H\"older-Zygmund spaces, where the weights are regularly varying functions. The analysis of these spaces is carried out via the wavelet transform and generalized Littlewood-Paley pairs.Comment: 18 page