Lie Group and Algebraic Groups: (Springer Series in Soviet Mathematics)

Focusing on Lie groups and algebraic groups, this work was created from hectographed notes in Russian from Moscow University. It has been extended and organized to develop the material through the posing of problems and to illustrate it through a wealth of examples. Tables cover decomposition of representations into irreducible components, which should be useful to physicists. Topics covered include: the correspondence between Lie groups and Lie algebras; elements of algebraic geometry and of algebraic group theory over fields of real and complex numbers; and the main facts of the theory of semi-simple Lie groups (real and complex, their local and global classification included) and their representations.

Table of Contents:
1. Lie Groups.- 1. Background.- 1 . Lie Groups.- 2 . Lie Subgroups.- 3 . Homomorphisms, Linear Representations and Actions of Lie Groups.- 4 . Operations on Linear Representations.- 5 . Orbits and Stabilizers.- 6 . The Image and the Kernel of a Homomorphism.- 7 . Coset Manifolds and Quotient Groups.- 8 . Theorems on Transitive Actions and Epimorphisms.- 9 . Homogeneous Spaces.- 10 . Inverse Image of a Lie Subgroup with Respect to a Homomorphism.- 11 . Semidirect Product.- Exercises.- Hints to Problems.- 2. Tangent Algebra.- 1 . Definition of the Tangent Algebra.- 2 . Tangent Homomorphism.- 3 . The Tangent Algebra of a Stabilizer.- 4 . The Adjoint Representation and the Jacobi Identity.- 5 . Differential Equations for Paths on a Lie Group.- 6 . Uniqueness Theorem for Lie Group Homomorphisms.- 7 . Exponential Map.- 8 . Existence Theorem for Lie Group Homomorphisms.- 9 . Virtual Lie Subgroups.- 10 . Automorphisms and Derivations.- 11 . The Tangent Algebra of a Semidirect Product of Lie Groups.- Exercises.- Hints to Problems.- 3. Connectedness and Simple Connectedness.- 1 . Connectedness.- 2 . Covering Homomorphisms.- 3 . Simply Connected Covering Lie Groups.- 4 . Exact Homotopy Sequence.- Exercises.- Hints to Problems.- 4. The Derived Algebra and the Radical.- 1 . The Commutator Group and the Derived Algebra.- 2 . Malcev Closures.- 3 . Existence of Virtual Lie Subgroups.- 4 . Solvable Lie Groups.- 5 . Lie's Theorem.- 6 . The Radical. Semisimple Lie Groups.- 7 . Complexification.- Exercises.- Hints to Problems.- 2. Algebraic Varieties.- 1. Affine Algebraic Varieties.- 1 . Embedded Affine Varieties.- 2 . Morphisms.- 3 . Zariski Topology.- 4 . The Direct Product.- 5 . Homomorphism Extension Theorems.- 6 . The Image of a Dominant Morphism.- 7 . Hilbert's Nullstellensatz.- 8 . Rational Functions.- 9 . Rational Maps.- 10 . Factorization of a Morphism.- Exercises.- Hints to Problems.- 2. Projective and Quasiprojective Varieties.- 1 . Graded Algebras.- 2 . Embedded Projective Algebraic Varieties.- 3 . Sheaves of Functions.- 4 . Sheaves of Algebras of Rational Functions.- 5 . Quasiprojective Varieties.- 6 . The Direct Product.- 7 . Flag Varieties.- Exercises.- Hints to Problems.- 3. Dimension and Analytic Properties of Algebraic Varieties.- 1 . Definition of the Dimension and its Main Properties.- 2 . Derivations of the Algebra of Functions.- 3 . Simple Points.- 4 . The Analytic Structure of Complex and Real Algebraic Varieties.- 5 . Realification of Complex Algebraic Varieties.- 6 . Forms of Vector Spaces and Algebras.- 7 . Real Forms of Complex Algebraic Varieties.- Exercises.- Hints to Problems.- 3. Algebraic Groups.- 1. Background.- 1 . Main Definitions.- 2 . Complex and Real Algebraic Groups.- 3 . Semidirect Products.- 4 . Certain Theorems on Subgroups and Homomorphisms of Algebraic Groups.- 5 . Actions of Algebraic Groups.- 6 . Existence of a Faithful Linear Representation.- 7 . The Coset Variety and the Quotient Group.- Exercises.- Hints to Problems.- 2. Commutative and Solvable Algebraic Groups.- 1 . The Jordan Decomposition of a Linear Operator.- 2 . Commutative Unipotent Algebraic Linear Groups.- 3 . Algebraic Tori and Quasitori.- 4 . The Jordan Decomposition in an Algebraic Group.- 5 . The Structure of Commutative Algebraic Groups.- 6 . Borel's Theorem.- 7 . The Splitting of a Solvable Algebraic Group.- 8 . Semisimple Elements of a Solvable Algebraic Group.- 9 . Borel Subgroups.- Exercises.- Hints of Problems.- 3. The Tangent Algebra.- 1 . Connectedness of Irreducible Complex Algebraic Groups.- 2 . The Rational Structure on the Tangent Algebra of a Torus.- 3 . Algebraic Subalgebras.- 4 . The Algebraic Structure on Certain Complex Lie Groups.- 5 . Engel's Theorem.- 6 . Unipotent Algebraic Linear Groups.- 7 . The Jordan Decomposition in the Tangent Algebra of an Algebraic Group.- 8 . The Tangent Algebra of a Real Algebraic Group.- 9 . Inverse Image of a Lie Subgroup with Respect to a Homomorphism.- 11 . Semidirect Product.- Exercises.- Hints to Problems.- 2. Tangent Algebra.- 1 . Definition of the Tangent Algebra.- 2 . Tangent Homomorphism.- 3 . The Tangent Algebra of a Stabilizer.- 4 . The Adjoint Representation and the Jacobi Identity.- 5 . Differential Equations for Paths on a Lie Group.- 6 . Uniqueness Theorem for Lie Group Homomorphisms.- 7 . Exponential Map.- 8 . Existence Theorem for Lie Group Homomorphisms.- 9 . Virtual Lie Subgroups.- 10 . Automorphisms and Derivations.- 11 . The Tangent Algebra of a Semidirect Product of Lie Groups.- Exercises.- Hints to Problems.- 3. Connectedness and Simple Connectedness.- 1 . Connectedness.- 2 . Covering Homomorphisms.- 3 . Simply Connected Covering Lie Groups.- 4 . Exact Homotopy Sequence.- Exercises.- Hints to Problems.- 4. The Derived Algebra and the Radical.- 1 . The Commutator Group and the Derived Algebra.- 2 . Malcev Closures.- 3 . Existence of Virtual Lie Subgroups.- 4 . Solvable Lie Groups.- 5 . Lie's Theorem.- 6 . The Radical. Semisimple Lie Groups.- 7 . Complexification.- Exercises.- Hints to Problems.- 2. Algebraic Varieties.- 1. Affine Algebraic Varieties.- 1 . Embedded Affine Varieties.- 2 . Morphisms.- 3 . Zariski Topology.- 4 . The Direct Product.- 5 . Homomorphism Extension Theorems.- 6 . The Image of a Dominant Morphism.- 7 . Hilbert's Nullstellensatz.- 8 . Rational Functions.- 9 . Rational Maps.- 10 . Factorization of a Morphism.- Exercises.- Hints to Problems.- 2. Projective and Quasiprojective Varieties.- 1 . Graded Algebras.- 2 . Embedded Projective Algebraic Varieties.- 3 . Sheaves of Functions.- 4 . Sheaves of Algebras of Rational Functions.- 5 . Quasiprojective Varieties.- 6 . The Direct Product.- 7 . Flag Varieties.- Exercises.- Hints to Problems.- 3. Dimension and Analytic Properties of Algebraic Varieties.- 1 . Definition of the Dimension and its Main Properties.- 2 . Derivations of the Algebra of Functions.- 3 . Simple Points.- 4 . The Analytic Structure of Complex and Real Algebraic Varieties.- 5 . Realification of Complex Algebraic Varieties.- 6 . Forms of Vector Spaces and Algebras.- 7 . Real Forms of Complex Algebraic Varieties.- Exercises.- Hints to Problems.- 3. Algebraic Groups.- 1. Background.- 1 . Main Definitions.- 2 . Complex and Real Algebraic Groups.- 3 . Semidirect Products.- 4 . Certain Theorems on Subgroups and Homomorphisms of Algebraic Groups.- 5 . Actions of Algebraic Groups.- 6 . Existence of a Faithful Linear Representation.- 7 . The Coset Variety and the Quotient Group.- Exercises.- Hints to Problems.- 2. Commutative and Solvable Algebraic Groups.- 1 . The Jordan Decomposition of a Linear Operator.- 2 . Commutative Unipotent Algebraic Linear Groups.- 3 . Algebraic Tori and Quasitori.- 4 . The Jordan Decomposition in an Algebraic Group.- 5 . The Structure of Commutative Algebraic Groups.- 6 . Borel's Theorem.- 7 . The Splitting of a Solvable Algebraic Group.- 8 . Semisimple Elements of a Solvable Algebraic Group.- 9 . Borel Subgroups.- Exercises.- Hints of Problems.- 3. The Tangent Algebra.- 1 . Connectedness of Irreducible Complex Algebraic Groups.- 2 . The Rational Structure on the Tangent Algebra of a Torus.- 3 . Algebraic Subalgebras.- 4 . The Algebraic Structure on Certain Complex Lie Groups.- 5 . Engel's Theorem.- 6 . Unipotent Algebraic Linear Groups.- 7 . The Jordan Decomposition in the Tangent Algebra of an Algebraic Group.- 8 . The Tangent Algebra of a Real Algebraic Group.- 9 . The Union of Borel Subgroups and the Centralizers of Tori.- Exercises.- Hints to Problems.- 4. Compact Linear Groups.- 1 . A Fixed Point Theorem.- 2 . Complete Reducibility.- 3 . Separating Orbits with the Help of Invariants.- 4 . Algebraicity.- Exercises.- Hints to Problems.- 4. Complex Semisimple Lie Groups.- 1. Preliminaries.- 1 . Invariant Scalar Products.- 2 . Algebraicity.- 3 . Normal Subgroups.- 4 . Weight and Root Decompositions.- 5 . Root Decompositions and Root Systems of Classical Lie Algebras.- 6 . Three-Dimensional Subalgebras.- Exercises.- Hints to Problems.- 2. Root Systems.- 1 . Principal Definitions and Examples.- 2 . Weyl Chambers and Simple Roots.- 3 . Borel Subgroups and Maximal Tori.- 4 . Weyl Group.- 5 . Dynkin Diagrams.- 6 . Cartan Matrices.- 7 . Classification.- 8 . Root and Weight Lattices.- Exercises.- Hints to Problems.- 3. Existence and Uniqueness Theorems.- 1 . Free Lie Algebras, Generators and Defining Relations.- 2 . Uniqueness Theorems.- 3 . Existence Theorems.- 4 . The Linearity of a Connected Complex Semisimple Lie Group.- 5 . The Center and the Fundamental Group.- 6 . Classification of Connected Semisimple Lie Groups.- 7 . Classification of Irreducible Representations.- Exercises.- Hints to Problems.- 4. Automorphisms.- 1 . The Group of Outer Automorphisms.- 2 . Semisimple Automorphisms.- 3 . Characters and Automorphisms of Quasi-Tori.- 4 . Affine Root Decomposition.- 5 . Affine Weyl Group.- 6 . Affine Roots of a Simple Lie Algebra.- 7 . Classification of Unitary Automorphisms of Simple Lie Algebras. 212 8 . Fixed Points of Semisimple Automorphisms of a Simply Connected Group.- Exercises.- Hints to Problems.- 5. Real Semisimple Lie Groups.- 1. Real Forms of Complex Semisimple Lie Groups and Algebras.- 1 . Real Structures and Real Forms.- 2 . Real Forms of Classical Lie Groups and Algebras.- 3 . The Compact Real Form.- 4 . Real Forms and Involutive Automorphisms.- 5 . Involutive Automorphisms of Complex Simple Lie Algebras.- 6 . Classification of Real Simple Lie Algebras.- Exercises.- Hints to Problems.- 2. Compact Lie Groups and Reductive Algebraic Groups.- 1 . Polar Decomposition.- 2 . Lie Groups with Compact Tangent Algebras.- 3 . Compact Real Forms of Reductive Algebraic Groups.- 4 . Linearity of Compact Lie Groups.- 5 . Correspondence Between Compact Lie Groups and Reductive Algebraic Groups.- 6 . Complete Reducibility of Linear Representations.- 7 . Maximal Tori in Compact Lie Groups.- Exercises.- Hints to Problems.- 3. Cartan Decomposition.- 1 . Cartan Decomposition of a Semisimple Lie Algebra.- 2 . Cartan Decomposition of a Semisimple Lie Group.- 3 . Conjugacy of Maximal Compact Subgroups.- 4 . Canonically Embedded Subalgebras.- 5 . Classification of Connected Semisimple Lie Groups.- 6 . Linearizer.- Exercises.- Hints to Problems.- 4. Real Root Decomposition.- 1 . Maximal (R-Diagonalizable Subalgebras.- 2 . Real Root Systems.- 3 . Satake Diagram.- 4 . Split Semisimple Lie Algebras.- 5 . Iwasawa Decomposition.- Exercises.- Hints to Problems.- 6. Levi Decomposition.- 1 . Levi's Theorem.- 2 . Existence of a Lie Group with the Given Tangent Algebra.- 3 . Malcev's Theorem.- 4 . Algebraic Levi Decomposition.- Exercises.- Hints to Problems.- Reference Chapter.- 1. Useful Formulae.- 1 . Weyl Groups and Exponents.- 2 . Linear Representations of Complex Semisimple Lie Algebras.- 3 . Linear Representations of Real Semisimple Lie Algebras.- . Inverse Image of a Lie Subgroup with Respect to a Homomorphism.- 11 . Semidirect Product.- Exercises.- Hints to Problems.- 2. Tangent Algebra.- 1 . Definition of the Tangent Algebra.- 2 . Tangent Homomorphism.- 3 . The Tangent Algebra of a Stabilizer.- 4 . The Adjoint Representation and the Jacobi Identity.- 5 . Differential Equations for Paths on a Lie Group.- 6 . Uniqueness Theorem for Lie Group Homomorphisms.- 7 . Exponential Map.- 8 . Existence Theorem for Lie Group Homomorphisms.- 9 . Virtual Lie Subgroups.- 10 . Automorphisms and Derivations.- 11 . The Tangent Algebra of a Semidirect Product of Lie Groups.- Exercises.- Hints to Problems.- 3. Connectedness and Simple Connectedness.- 1 . Connectedness.- 2 . Covering Homomorphisms.- 3 . Simply Connected Covering Lie Groups.- 4 . Exact Homotopy Sequence.- Exercises.- Hints to Problems.- 4. The Derived Algebra and the Radical.- 1 . The Commutator Group and the Derived Algebra.- 2 . Malcev Closures.- 3 . Existence of Virtual Lie Subgroups.- 4 . Solvable Lie Groups.- 5 . Lie's Theorem.- 6 . The Radical. Semisimple Lie Groups.- 7 . Complexification.- Exercises.- Hints to Problems.- 2. Algebraic Varieties.- 1. Affine Algebraic Varieties.- 1 . Embedded Affine Varieties.- 2 . Morphisms.- 3 . Zariski Topology.- 4 . The Direct Product.- 5 . Homomorphism Extension Theorems.- 6 . The Image of a Dominant Morphism.- 7 . Hilbert's Nullstellensatz.- 8 . Rational Functions.- 9 . Rational Maps.- 10 . Factorization of a Morphism.- Exercises.- Hints to Problems.- 2. Projective and Quasiprojective Varieties.- 1 . Graded Algebras.- 2 . Embedded Projective Algebraic Varieties.- 3 . Sheaves of Functions.- 4 . Sheaves of Algebras of Rational Functions.- 5 . Quasiprojective Varieties.- 6 . The Direct Product.- 7 . Flag Varieties.- Exercises.- Hints to Problems.- 3. Dimension and Analytic Properties of Algebraic Varieties.- 1 . Definition of the Dimension and its Main Properties.- 2 . Derivations of the Algebra of Functions.- 3 . Simple Points.- 4 . The Analytic Structure of Complex and Real Algebraic Varieties.- 5 . Realification of Complex Algebraic Varieties.- 6 . Forms of Vector Spaces and Algebras.- 7 . Real Forms of Complex Algebraic Varieties.- Exercises.- Hints to Problems.- 3. Algebraic Groups.- 1. Background.- 1 . Main Definitions.- 2 . Complex and Real Algebraic Groups.- 3 . Semidirect Products.- 4 . Certain Theorems on Subgroups and Homomorphisms of Algebraic Groups.- 5 . Actions of Algebraic Groups.- 6 . Existence of a Faithful Linear Representation.- 7 . The Coset Variety and the Quotient Group.- Exercises.- Hints to Problems.- 2. Commutative and Solvable Algebraic Groups.- 1 . The Jordan Decomposition of a Linear Operator.- 2 . Commutative Unipotent Algebraic Linear Groups.- 3 . Algebraic Tori and Quasitori.- 4 . The Jordan Decomposition in an Algebraic Group.- 5 . The Structure of Commutative Algebraic Groups.- 6 . Borel's Theorem.- 7 . The Splitting of a Solvable Algebraic Group.- 8 . Semisimple Elements of a Solvable Algebraic Group.- 9 . Borel Subgroups.- Exercises.- Hints of Problems.- 3. The Tangent Algebra.- 1 . Connectedness of Irreducible Complex Algebraic Groups.- 2 . The Rational Structure on the Tangent Algebra of a Torus.- 3 . Algebraic Subalgebras.- 4 . The Algebraic Structure on Certain Complex Lie Groups.- 5 . Engel's Theorem.- 6 . Unipotent Algebraic Linear Groups.- 7 . The Jordan Decomposition in the Tangent Algebra of an Algebraic Group.- 8 . The Tangent Algebra of a Real Algebraic Group.- 9 . The Union of Borel Subgroups and the Centralizers of Tori.- Exercises.- Hints to Problems.- 4. Compact Linear Groups.- 1 . A Fixed Point Theorem.- 2 . Complete Reducibility.- 3 . Separating Orbits with the Help of Invariants.- 4 . Algebraicity.- Exercises.- Hints to Problems.- 4. Complex Semisimple Lie Groups.- 1. Preliminaries.- 1 . Invariant Scalar Products.- 2 . Algebraicity.- 3 . Normal Subgroups.- 4 . Weight and Root Decompositions.- 5 . Root Decompositions and Root Systems of Classical Lie Algebras.- 6 . Three-Dimensional Subalgebras.- Exercises.- Hints to Problems.- 2. Root Systems.- 1 . Principal Definitions and Examples.- 2 . Weyl Chambers and Simple Roots.- 3 . Borel Subgroups and Maximal Tori.- 4 . Weyl Group.- 5 . Dynkin Diagrams.- 6 . Cartan Matrices.- 7 . Classification.- 8 . Root and Weight Lattices.- Exercises.- Hints to Problems.- 3. Existence and Uniqueness Theorems.- 1 . Free Lie Algebras, Generators and Defining Relations.- 2 . Uniqueness Theorems.- 3 . Existence Theorems.- 4 . The Linearity of a Connected Complex Semisimple Lie Group.- 5 . The Center and the Fundamental Group.- 6 . Classification of Connected Semisimple Lie Groups.- 7 . Classification of Irreducible Representations.- Exercises.- Hints to Problems.- 4. Automorphisms.- 1 . The Group of Outer Automorphisms.- 2 . Semisimple Automorphisms.- 3 . Characters and Automorphisms of Quasi-Tori.- 4 . Affine Root Decomposition.- 5 . Affine Weyl Group.- 6 . Affine Roots of a Simple Lie Algebra.- 7 . Classification of Unitary Automorphisms of Simple Lie Algebras. 212 8 . Fixed Points of Semisimple Automorphisms of a Simply Connected Group.- Exercises.- Hints to Problems.- 5. Real Semisimple Lie Groups.- 1. Real Forms of Complex Semisimple Lie Groups and Algebras.- 1 . Real Structures and Real Forms.- 2 . Real Forms of Classical Lie Groups and Algebras.- 3 . The Compact Real Form.- 4 . Real Forms and Involutive Automorphisms.- 5 . Involutive Automorphisms of Complex Simple Lie Algebras.- 6 . Classification of Real Simple Lie Algebras.- Exercises.- Hints to Problems.- 2. Compact Lie Groups and Reductive Algebraic Groups.- 1 . Polar Decomposition.- 2 . Lie Groups with Compact Tangent Algebras.- 3 . Compact Real Forms of Reductive Algebraic Groups.- 4 . Linearity of Compact Lie Groups.- 5 . Correspondence Between Compact Lie Groups and Reductive Algebraic Groups.- 6 . Complete Reducibility of Linear Representations.- 7 . Maximal Tori in Compact Lie Groups.- Exercises.- Hints to Problems.- 3. Cartan Decomposition.- 1 . Cartan Decomposition of a Semisimple Lie Algebra.- 2 . Cartan Decomposition of a Semisimple Lie Group.- 3 . Conjugacy of Maximal Compact Subgroups.- 4 . Canonically Embedded Subalgebras.- 5 . Classification of Connected Semisimple Lie Groups.- 6 . Linearizer.- Exercises.- Hints to Problems.- 4. Real Root Decomposition.- 1 . Maximal (R-Diagonalizable Subalgebras.- 2 . Real Root Systems.- 3 . Satake Diagram.- 4 . Split Semisimple Lie Algebras.- 5 . Iwasawa Decomposition.- Exercises.- Hints to Problems.- 6. Levi Decomposition.- 1 . Levi's Theorem.- 2 . Existence of a Lie Group with the Given Tangent Algebra.- 3 . Malcev's Theorem.- 4 . Algebraic Levi Decomposition.- Exercises.- Hints to Problems.- Reference Chapter.- 1. Useful Formulae.- 1 . Weyl Groups and Exponents.- 2 . Linear Representations of Complex Semisimple Lie Algebras.- 3 . Linear Representations of Real Semisimple Lie Algebras.- 2. Tables.- Table 1. Weights and Roots.- Table 2. Matrices Inverse to Cartan Matrices.- Table 3. Centers, Outer Automorphisms and Bilinear Invariants.- Table 4. Exponents.- Table 5. Decomposition of Tensor Products and Dimensions of Certain Representations.- Table 6. Affine Dynkin Diagrams.- Table 7. Involutive Automorphisms of Complex Simple Lie Algebras.- Table 8. Matrix Realizations of Classical Real Lie Algebras.- Table 9. Real Simple Lie Algebras.- Table 10. Centers and Linearizers of Simply Connected Real Simple Lie Groups.