Introduction to Quantum Algebras

Abstract

The concept of a quantum algebra is made easy through the investigation of the prototype algebras $u_{qp}(2)$, $su_q(2)$ and $u_{qp}(1,1)$. The latter quantum algebras are introduced as deformations of the corresponding Lie algebras~; this is achieved in a simple way by means of $qp$-bosons. The Hopf algebraic structure of $u_{qp}(2)$ is also discussed. The basic ingredients for the representation theory of $u_{qp}(2)$ are given. Finally, in connection with the quantum algebra $u_{qp}(2)$, we discuss the $qp$-analogues of the harmonic oscillator and of the (spherical and hyperbolical) angular momenta.