Quantum algebras are a mathematical tool which provides us with a class of symmetries wider than that of Lie algebras, which are contained in the former as a special case. After a self-contained introduction to the necessary mathematical tools (q-numbers, q-analysis, q-oscillators, q-algebras), the suq(2) rotator model and its extensions, the construction of deformed exactly soluble models (Int...

We prove in this paper a noncommutative version of Leray Theorem and then Leray-Serre Spectral Theorem for noncommutative Serre fibrations: for NC Serre fibration there are converging spectral sequences with E2 terms as E2 p,q = HPp(A;HPq(B,A)) =⇒ HPp+q(B) and E 2 p,q = HPp(A;Kq(B,A)) =⇒ Kp+q(B). Introduction The ideas of using spectral sequences to operator algebras was started in [C]. In that...

We show that for q 6= −1 the q-graded tensor product fails to preserve the q-differential structure of the product algebra and therefore there is no natural tensor product construction for q-differential algebras.

A basis of symmetric functions, which we denote by q*(X; q, t), was introduced in the work of Ram and King and Wybourne in order to describe the irreducible characters of the Hecke algebras of type A. In this work we give combinatorial descriptions of the expansions of the functions q*(X; q, t) in terms of the classical bases of symmetric functions and apply these results in determining the det...

This definition goes back to J.Dixmier, see [Di, 11.1.6]. He considered index because of its importance in Representation Theory. The problem of computing the index may also be treated as part of Invariant Theory. For, if q is an algebraic Lie algebra and Q is a corresponding algebraic group, then ind q equals the transcendence degree of the field of Q-invariant rational functions on q. If q is...

It is shown that q-deformed quantummechanics (systems with q-deformed Heisenberg commutation relations) can be interpreted as an ordinary quantum mechanics on Kähler manifolds, or as a quantum theory with second (or first)-class constraints. 1. The q-deformed Heisenberg-Weyl algebras [1], [2] exhibiting the quantum group symmetries [3],[4] have attracted much attention of physicists and mathema...