Theory of Lie Groups

This book collects a series of contributions addressing the various contexts in which the theory of Lie groups is applied. A preliminary chapter serves the reader both as a basic reference source and as an ongoing thread that runs through the subsequent chapters. From representation theory and Gerstenhaber algebras to control theory, from differential equations to Finsler geometry and Lepage manifolds, the book introduces young researchers in Mathematics to a wealth of different topics, encouraging a multidisciplinary approach to research. As such, it is suitable for students in doctoral courses, and will also benefit researchers who want to expand their field of interest.

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 .