Introduction

Abstract

We will present an original approach to probability theory through operator algebras — representations of Lie algebras. Along the way we will meet ma ny special functions, particularly orthogonal polynomials. Principal roles will be played by the Heisenberg-Weyl algebra and the sl(2) algebra. The oscillator algebra will appear as well. These are introduced in Chapter 1 where basic techniques for working with noncommutative variables are presented. Formulas for the groups corresponding to the various algebras are discussed there. Chapter 2 is a brief foray into the subject of hypergeometric functions. Here some illustrations of our algebraic approach are to be found. Chapter 3 is a presentation of probability theory from the operator algebra point of view —often referred to as ‘quantum probability’. The results are stated in the finite-dimensional case, but they are formulated (and later used) in the infinite-dimensional case, so that the presentation is applicable for the general theory. Fock spaces are introduced and connections with analytic structure, e.g. reproducing kernels, are given. Clebsch-Gordan coefficients are calculated, with connections to orthogonal polynomials.