Noncompact Semisimple Lie Algebras and Groups by Dobrev, Vladimir K. -- Read -- Imperial Library of Trantor

Index

Cover Title Copyright Contents 1 Introduction

1.1 Symmetries 1.2 Invariant Differential Operators 1.3 Sketch of Procedure 1.4 Organization of the Book

2 Lie Algebras and Groups

2.1 Generalities on Lie Algebras

2.1.1 Lie Algebras 2.1.2 Subalgebras, Ideals, and Factor-Algebras 2.1.3 Representations 2.1.4 Solvable Lie Algebras 2.1.5 Nilpotent Lie Algebras 2.1.6 Semisimple Lie Algebras 2.1.7 Examples

2.2 Elements of Group Theory

2.2.1 Definition of a Group 2.2.2 Group Actions 2.2.3 Subgroups and Factor-Groups 2.2.4 Homomorphisms 2.2.5 Direct and Semidirect Products of Groups

2.3 Structure of Semisimple Lie Algebras

2.3.1 Cartan Subalgebra 2.3.2 Lemmas on Root Systems 2.3.3 Weyl Group 2.3.4 Cartan Matrix

2.4 Classification of Kac–Moody Algebras 2.5 Realization of Semisimple Lie Algebras

2.5.1 Special Linear Algebra 2.5.2 Odd Orthogonal Lie Algebra 2.5.3 Symplectic Lie Algebra 2.5.4 Even Orthogonal Lie Algebra 2.5.5 Exceptional Lie Algebra G2 2.5.6 Exceptional Lie Algebra F4 2.5.7 Exceptional Lie Algebras Eℓ

2.6 Realization of Affine Kac–Moody Algebras

2.6.1 Realization of Affine Type 1 Kac–Moody Algebras 2.6.2 Realization of Affine Type 2 and 3 Kac–Moody Algebras 2.6.3 Root System for the Algebras AFF 2 & 3

2.7 Chevalley Generators, Serre Relations, and Cartan–Weyl Basis 2.8 Highest Weight Representations of Kac–Moody Algebras 2.9 Verma Modules 2.10 Irreducible Representations

2.10.1 Aℓ 2.10.2 Cℓ 2.10.3 Bℓ 2.10.4 Dℓ 2.10.5 Eℓ 2.10.6 F4 2.10.7 G2

2.11 Characters of Highest Weight Modules

2.11.1 Irreducible Quotients of Reducible Verma Modules 2.11.2 Embedding Patterns and Mulltiplets 2.11.3 Characters of Generic Highest Weight Modules 2.11.4 Characters for Nondominant Weights 2.11.5 Characters in the Affine Case 2.11.6 Example of A1(1)

3 Real Semisimple Lie Algebras

3.1 Structure of Noncompact Semisimple Lie Algebras

3.1.1 Preliminaries 3.1.2 The Structure in Detail

3.2 Classification of Noncompact Semisimple Lie Algebras 3.3 Parabolic Subalgebras 3.4 Complex Simple Lie Algebras Considered as Real Lie Algebras 3.5 AI : SL(n, R) 3.6 AII : SU∗(2n) 3.7 AIII : SU(p, r)

3.7.1 Case SU(n, n), n > 1 3.7.2 Case SU(p, r), p > r ≥ 1

3.8 BDI : SO(p, r) 3.9 CI : Sp(n, R), n > 1 3.10 CII : Sp(p, r) 3.11 DIII : SO∗(2n) 3.12 Real Forms of the Exceptional Simple Lie Algebras

3.12.1 EI : E′6 3.12.2 EII : E′′6 3.12.3 EIII : E′′′6 3.12.4 EIV : Eiv6 3.12.5 EV : E′7 3.12.6 EVI : E′′7 3.12.7 EVII : E′′′7 3.12.8 EVIII : E′8 3.12.9 EIX : E′′8 3.12.10 FI : F′4 3.12.11 FII : F′′4 3.12.12 GI: G′2

4 Invariant Differential Operators

4.1 Lie Groups

4.1.1 Preliminaries 4.1.2 Classical Groups 4.1.3 Types of Lie Groups 4.1.4 Cartan Subgroups 4.1.5 Cartan and Iwasawa Decompositions 4.1.6 Parabolic Subgroups

4.2 Preliminaries on Group Representation Theory

4.2.1 Representations and Modules 4.2.2 Reducibility and Irreducibility 4.2.3 Operations on Representations 4.2.4 Induced Representations

4.3 Elementary Representations

4.3.1 Compact Lie Groups 4.3.2 Noncompact Lie Groups 4.3.3 Knapp–Stein Integral Operators 4.3.4 ERs of Complex Lie Groups

4.4 Unitary Irreducible Representations 4.5 Associated Verma Modules 4.6 Invariant Differential Operators

4.6.1 Canonical Construction 4.6.2 Multiplets of GVMs and ERs

4.7 Example of SL(2,R)

4.7.1 Elementary Representations 4.7.2 Discrete Series and Limits Thereof 4.7.3 Positive Energy Representations

4.8 Explicit Formulae for Singular Vectors

4.8.1 Aℓ 4.8.2 Dℓ 4.8.3 Eℓ 4.8.4 Bℓ 4.8.5 Cℓ 4.8.6 F4 4.8.7 G2 4.8.8 Nonstraight Roots

5 Case of the Anti-de Sitter Group

5.1 Preliminaries

5.1.1 Lie Algebra 5.1.2 Finite-Dimensional Realization 5.1.3 Structure Theory 5.1.4 Lie Groups

5.2 Representations and Invariant Operators

5.2.1 Elementary Representations 5.2.2 Elementary Representations Induced from P0 5.2.3 Singular Vectors 5.2.4 Invariant Differential Operators 5.2.5 Reducible ERs 5.2.6 Holomorphic Discrete Series and Positive Energy Representations 5.2.7 Invariant Differential Operators and Equations Related to Positive Energy UIRs 5.2.8 Rac 5.2.9 Di 5.2.10 Massless Representations

5.3 Classification of so(5, C) Verma Modules and P0-Induced ERs 5.4 Character Formulae

5.4.1 Character Formulae of AdS Irreps 5.4.2 Character Formulae of Positive Energy UIRs

6 Conformal Case in 4D

6.1 Preliminaries

6.1.1 Realizations of the Group SU(2, 2) 6.1.2 Lie Algebra of SU(2, 2) 6.1.3 Restricted Root System, Bruhat and Iwasawa Decompositions 6.1.4 Restricted Weyl Group W(G, A0) 6.1.5 Parabolic Subalgebras 6.1.6 Complexified Lie Algebra 6.1.7 Compact and Noncompact Roots 6.1.8 Important Subgroups of G

6.2 Elementary Representations of SU(2, 2)

6.2.1 ERs from the Minimal Parabolic Subgroup P0 6.2.2 ERs from the Maximal Cuspidal Parabolic Subgroup P1 6.2.3 ERs from the Maximal Noncuspidal Parabolic Subgroup P2 6.2.4 Noncompact Picture of the ERs 6.2.5 Properties of ERs 6.2.6 Integral Invariant Operators

6.3 Invariant Differential Operators and Multiplet Classification of the Reducible ERs

6.3.1 Explicit Expressions for the Invariant Differential Operators 6.3.2 Multiplet Classification: Case P0 6.3.3 Multiplet Classification: Case P2 6.3.4 Holomorphic Discrete Series and Lowest Weight Representations 6.3.5 Multiplet Classification: Case P1

7 Kazhdan–Lusztig Polynomials, Subsingular Vectors, and Conditionally Invariant Equations

7.1 Subsingular Vectors

7.1.1 Preliminaries 7.1.2 Definition 7.1.3 Bernstein–Gel’fand–Gel’fand Example 7.1.4 The Other Archetypal sl(4, C) Example

7.2 Kazhdan–Lusztig Polynomials 7.3 Characters of LWM and Nontrivial KL Polynomials

7.3.1 Preliminaries on Characters of Lowest Weight Modules 7.3.2 Case sl(4, C) 7.3.3 Related Character Formulae

7.4 KL Polynomials and Subsingular Vectors: A Conjecture 7.5 Conditionally Invariant Differential Equations

7.5.1 Preliminaries 7.5.2 Conditionally Invariant Operators

7.6 Application to sl(4, C)

7.6.1 Equations Arising from the BGG Example 7.6.2 Equations Arising from the Other Archetypal sl(4, C) Example

8 Invariant Differential Operators for Noncompact Lie Algebras Parabolically Related to Conformal Lie Algebras

8.1 Generalities 8.2 The Pseudo-Orthogonal Algebras so(p,q)

8.2.1 Choice of Parabolic Subalgebra 8.2.2 Main Multiplets 8.2.3 Reduced Multiplets and Their Representations 8.2.4 Conservation Laws for so(p,q) 8.2.5 Remarks on Shadow Fields and History 8.2.6 Case so(3, 3) = sl(4, R)

8.3 The Lie Algebra su(n,n) and Parabolically Related

8.3.1 Multiplets of su(3, 3) and sl(6, R) 8.3.2 Multiplets of su(4, 4), sl(8, R), and su∗(8)

8.4 Multiplets and Representations for sp(n, R) and sp(r, r)

8.4.1 Preliminaries 8.4.2 The Case sp(3, R) 8.4.3 The Case sp(4, R) and sp(2, 2) 8.4.4 The Case sp(5, R) 8.4.5 The Case sp(6, R) and sp(3, 3) 8.4.6 Summary for sp(n, R)

8.5 SO∗(4n) Case

8.5.1 Main Multiplets 8.5.2 Reduced Multiplets and Minimal Irreps

8.6 The Lie Algebras E7(–25) and E7(7)

8.6.1 Main Type of Multiplets 8.6.2 Reduced Multiplets

8.7 The Lie Algebras E6(–14), E6(6), and E6(2)

8.7.1 Main Type of Multiplets 8.7.2 Reduced Multiplets

9 Multilinear Invariant Differential Operators from New Generalized Verma Modules

9.1 Preliminaries 9.2 k-Verma Modules 9.3 Singular Vectors of k-Verma Modules

9.3.1 Definition 9.3.2 k = 2 9.3.3 k = 3

9.4 Multilinear Invariant Differential Operators 9.5 Bilinear Operators for SL(n, R) and SL(n, C)

9.5.1 Setting 9.5.2 Minimal Parabolic 9.5.3 SL(2, R) 9.5.4 SL(3, R)

9.6 Examples with k ≥ 3

Bibliography Author Index Subject Index

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