Springer Soviet Mathematics

1. Lie Groups.- x 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.- x2. 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.- x3. Connectedness and Simple Connectedness.- 1 . Connectedness.- 2 . Covering Homomorphisms.- 3 . Simply Connected Covering Lie Groups.- 4 . Exact Homotopy Sequence.- Exercises.- Hints to Problems.- x 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.- x1. 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.- x 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.- x 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.- x 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.- x2. 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.- x 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.- x4. 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.- x1. 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.- x2. 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.- x3. 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.- x4. 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.- x 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.- x 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.- x 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.- x 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.- x 1. Useful Formulae.- 1 . Weyl Groups and Exponents.- 2 . Linear Representations of Complex Semisimple Lie Algebras.- 3 . Linear Representations of Real Semisimple Lie Algebras.- x2. 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.