8 Explicit computations -- 9 Exercises -- 6: Algebraic groups: Jordan decomposition and applications -- 1 Introduction -- 2 The Jordan decomposition of a single operator -- 3 The Jordan decomposition of an algebra homomorphism and of a derivation -- 4 The Jordan decomposition for coalgebras -- 5 The Jordan decomposition for an affine algebraic group -- 6 Unipotency and semisimplicity -- 7 The solvable and the unipotent radical -- 8 Structure of solvable groups -- 9 The classical groups -- 9.1 The general linear group GLn -- 9.2 The special linear group SLn (case A) -- 9.3 The projective general linear group PGLn(k) (case A) -- 9.4 The special orthogonal group SOn (cases B, D) -- 9.5 The symplectic group Spn, n = 2m (case C) -- 10 Exercises -- 7: Actions of algebraic groups -- 1 Introduction -- 2 Actions: examples and first properties -- 3 Basic facts about the geometry of the orbits -- 4 Categorical and geometric quotients -- 5 Affinized quotients -- 6 The subalgebra of invariants -- 7 Induction and restriction of representations -- 8 Exercises -- 8: Homogeneous spaces -- 1 Introduction -- 2 Embedding H-modules inside G-modules -- 3 Definition of subgroups in terms of semi-invariants -- 4 The coset space G/H as a geometric quotient -- 5 Quotients by normal subgroups -- 6 Applications and examples -- 7 Exercises -- 9: Algebraic groups and Lie algebras in characteristic zero -- 1 Introduction -- 2 Correspondence between subgroups and subalgebras -- 3 Algebraic Lie algebras -- 4 Exercises -- 10: Reductivity -- 1 Introduction -- 2 Linear and geometric reductivity -- 3 Examples of linearly and geometrically reductive groups -- 4 Reductivity and the structure of the group -- 5 Reductive groups are linearly reductive in characteristic zero -- 6 Exercises -- 11: Observable subgroups of affine algebraic groups