Algebraic Structures and Operators Calculus - Volume III: Representations of Lie Groups

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 . . . . . . . . . . . . . . . .

List of contents

I. General remarks.- II. Notations.- III. Lie algebras: some basics.- 1 Operator calculus and Appell systems.- I. Boson calculus.- II. Holomorphic canonical calculus.- III. Canonical Appell systems.- 2 Representations of Lie groups.- I. Coordinates on Lie groups.- II. Dual representations.- III. Matrix elements.- IV. Induced representations and homogeneous spaces.- 3 General Appell systems.- I. Convolution and stochastic processes.- II. Stochastic processes on Lie groups.- III. Appell systems on Lie groups.- 4 Canonical systems in several variables.- I. Homogeneous spaces and Cartan decompositions.- II. Induced representation and coherent states.- III. Orthogonal polynomials in several variables.- 5 Algebras with discrete spectrum.- I. Calculus on groups: review of the theory.- II. Finite-difference algebra.- III. q-HW algebra and basic hypergeometric functions.- IV. su2 and Krawtchouk polynomials.- V. e2 and Lommel polynomials.- 6 Nilpotent and solvable algebras.- I. Heisenberg algebras.- II. Type-H Lie algebras.- III. Upper-triangular matrices.- IV. Affine and Euclidean algebras.- 7 Hermitian symmetric spaces.- I. Basic structures.- II. Space of rectangular matrices.- III. Space of skew-symmetric matrices.- IV. Space of symmetric matrices.- 8 Properties of matrix elements.- I. Addition formulas.- II. Recurrences.- III. Quotient representations and summation formulas.- 9 Symbolic computations.- I. Computing the pi-matrices.- II. Adjoint group.- III. Recursive computation of matrix elements.- IV. Symbolic computation of Appell systems.- MAPLE output and procedures.- References.

Product details

• Authors: Feinsilver, P Feinsilver, P. Feinsilver, René Schott
• Publisher: Springer Netherlands
• Languages: English
• Product format: Paperback / Softback
• Released: 25.07.2012
• EAN: 9789401065573
• ISBN: 978-94-0-106557-3
• No. of pages: 230
• Dimensions: 175 mm x 240 mm x 14 mm
• Illustrations: IX, 230 p.
• Series: Mathematics and Its Applications; Mathematics and Its Applications (closed); Mathematics and Its Applications
• Subject: Natural sciences, medicine, IT, technology > Mathematics > Analysis