**[Cr:4, Lc:3, Tt:1, Lb:0]**

- Affine and projective varieties, prevarieties, morphisms of prevarieties, products, varieties, regular functions, complete varieties, algebraic function fields and dimension of a variety.
- Notion of an algebraic group, examples: GLn, SLn, finite groups. Identity component of an algebraic group, algebraic subgroups and Hopf algebras, action of algebraic groups on a varieties, transporters, existence of closed orbits, translations of functions, linearisation of affine algebraic groups.
- Lie algebra of an algebraic group, homogeneous spaces, rational representations and a theorem of Chevalley.
- Semisimple and unipotent elements, Jordan-Chevalley decomposition in algebraic groups, structure of commutative algebraic groups.
- Solvable groups, nilpotent and unipotent algebraic groups, Lie-Kolchin theorem, structure of connected solvable groups, Borel’s fixed point theorem.

- James Humphreys, Linear Algebraic Groups, Springer New York, 2012.
- Armand Borel, Linear Algebraic Groups, Springer-Verlag, 1991.
- T. A. Springer, Linear Algebraic Groups, Modern Birkháuser Classics, Birkháuser, 2008.