V5A1. Vorlesung Linear Algebraic groups Nicolas Perrin, [email protected] In this lecture we shall present the basic theory of algebraic groups over any algebraically closed field. Algebraic groups are groups which are algebraic varieties and such that the multiplication and the inverse maps are morphisms of algebraic varieties. Algebraic groups play an important role in many areas of mathematics especially in geometry and representation theory. In these lecture we shall concentrate on affine algebraic groups. We shall deal with the followings notions: • the Lie algebra of an algebraic group; • quotient of an algebraic group by a closed subgroup, homogeneous spaces; • tori, solvable groups and Borel subgroups; • structure of reductive groups; • classification of semisimple algebraic groups; • representations of semisimple algebraic groups. If time permits we may also say a few words on non algebraically closed fields. Prerequisities. Basics on algebraic geometry (for example the lecture of S¨ onke Rollenske) will be needed even if we shall not use the full machinery of algebraic geometry. We shall for example review the theory of tangent spaces for algebraic varieties during the lecture. We will also use some basic facts on Lie algebras and probably use root systems but I will probably recall at least the basic definitions on that subject. A good reference for Lie algebras is [Bou71], for root systems, see [Ser66] while we refer to [Har77] for algebraic geometry. There are several good books on algebraic groups like [Bor69], [Hum75] of [Spi81].

References [Bor69]

Borel, A. Linear algebraic groups. Notes taken by Hyman Bass W. A. Benjamin, Inc., New YorkAmsterdam 1969.

[Bou71]

´ ements de math´ematique. Fasc. XXVI. Groupes et alg`ebres de Lie. Chapitre I: Bourbaki, N. El´ Alg`ebres de Lie. Seconde ´edition. Actualit´es Scientifiques et Industrielles, No. 1285 Hermann, Paris 1971.

[Har77]

Hartshorne, R. Algebraic geometry. Graduate Texts in Mathematics, No. 52. Springer-Verlag, New York-Heidelberg, 1977.

[Hum75]

Humphreys, J.E. Linear algebraic groups. Graduate Texts in Mathematics, No. 21. Springer-Verlag, New York-Heidelberg, 1975.

[Spi81]

Springer, T.A. Linear algebraic groups. Progress in Mathematics, 9. Birkhuser, Boston, Mass., 1981.

[Ser66]

Serre, Jean-Pierre Alg`ebres de Lie semi-simples complexes. W. A. Benjamin, inc., New YorkAmsterdam 1966.

References [Bor69]

Borel, A. Linear algebraic groups. Notes taken by Hyman Bass W. A. Benjamin, Inc., New YorkAmsterdam 1969.

[Bou71]

´ ements de math´ematique. Fasc. XXVI. Groupes et alg`ebres de Lie. Chapitre I: Bourbaki, N. El´ Alg`ebres de Lie. Seconde ´edition. Actualit´es Scientifiques et Industrielles, No. 1285 Hermann, Paris 1971.

[Har77]

Hartshorne, R. Algebraic geometry. Graduate Texts in Mathematics, No. 52. Springer-Verlag, New York-Heidelberg, 1977.

[Hum75]

Humphreys, J.E. Linear algebraic groups. Graduate Texts in Mathematics, No. 21. Springer-Verlag, New York-Heidelberg, 1975.

[Spi81]

Springer, T.A. Linear algebraic groups. Progress in Mathematics, 9. Birkhuser, Boston, Mass., 1981.

[Ser66]

Serre, Jean-Pierre Alg`ebres de Lie semi-simples complexes. W. A. Benjamin, inc., New YorkAmsterdam 1966.