Lie Groups and Algebraic Groups (Paperback)

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°. The U