Algebraic Structures and Operators Calculus

Introduction I. General remarks ...1 II. Notations ...5 III. Lie algebras: some basics ...8 Chapter 1 Operator calculus and Appell systems I. Boson calculus ...17 II. Holomorphic canonical calculus ...18 III. Canonical Appell systems ...23 Chapter 2 Representations of Lie groups I. Coordinates on Lie groups ...28 II. Dual representations ...29 III. Matrix elements ...37 IV. Induced representations and homogeneous spaces ...40 General Appell systems Chapter 3 I. Convolution and stochastic processes ...44 II. Stochastic processes on Lie groups ...46 III. Appell systems on Lie groups ...49 Chapter 4 Canonical systems in several variables I. Homogeneous spaces and Cartan decompositions ...54 II. Induced representation and coherent states ...62 III. Orthogonal polynomials in several variables ...68 Chapter 5 Algebras with discrete spectrum I. Calculus on groups: review of the theory ...83 II. Finite-difference algebra ...85 III. q-HW algebra and basic hypergeometric functions ...89 IV. su2 and Krawtchouk polynomials ...93 V. e2 and Lommel polynomials ...101 Chapter 6 Nilpotent and solvable algebras I. Heisenberg algebras ...113 II. Type-H Lie algebras ...118 Vll III. Upper-triangular matrices ...125 IV. Affine and Euclidean algebras ...127 Chapter 7 Hermitian symmetric spaces I. Basic structures ...131 II. Space of rectangular matrices ...133 III. Space of skew-symmetric matrices ...136 IV. Space of symmetric matrices ...143 Chapter 8 Properties of matrix elements I. Addition formulas ...147 II. Recurrences ...148 III. Quotient representations and summation formulas ...149 Chapter 9 Symbolic computations I. Computing the pi-matrices ...153 II. Adjoint group ...154 III. Recursive computation of matrix elements ...