Results on topological spaces -- Rings and modules -- Integral extensions -- Factorial rings -- Field extensions -- Finitely generated algebras -- Gradings and filtrations -- Inductive limits -- Sheaves of functions -- Jordan decomposition and some basic results on groups -- Algebraic sets -- Prevarieties and varieties -- Projective varieties -- Dimension -- Morphisms and dimension -- Tangent spaces -- Normal varieties -- Root systems -- Lie algebras -- Semisimple and reductive Lie algebras -- Algebraic groups -- Affine algebraic groups -- Lie algebra of an algebraic group -- Correspondence between groups and Lie algebras -- Homogeneous spaces and quotients -- Solvable groups -- Reductive groups -- Borel subgroups, parabolic subgroups, Cartan subgroups -- Cartan subalgebras, Borel subalgebras and parabolic subalgebras -- Representations of semisimple Lie algebras -- Symmetric invariants -- S-triples -- Polarizations -- Results on orbits -- Centralizers -- ?-root systems -- Symmetric Lie algebras -- Semisimple symmetric Lie algebras -- Sheets of Lie algebras -- Index and linear forms.

The theory of Lie algebras and algebraic groups has been an area of active research in the last 50 years. It intervenes in many different areas of mathematics: for example invariant theory, Poisson geometry, harmonic analysis, mathematical physics. The aim of this book is to assemble in a single volume the algebraic aspects of the theory so as to present the foundation of the theory in characteristic zero. Detailed proofs are included and some recent results are discussed in the last chapters. All the prerequisites on commutative algebra and algebraic geometry are included.