Instructor: Amit Kumar Singh

Books:

1. Linear Algebraic Groups, by J. E. Humphreys.

2. Linear Algebraic Groups, by A. Borel

Lecture notes: Available here

Topics covered

Lecture 1: Affine varieties, projective varieties.

Lecture 2: Morphisms of varieties.

Lecture 3: The basic notions of algebraic groups

Suggested reading: Humphreys, section 7.1, 7.2.

Lecture 4: The identity component of algebraic groups, homomorphisms of algebraic groups.

Suggested reading: Humphreys, section 7.3, 7.4.

Assignment 1 (due Aug 21, 2023)

Lecture 5: Actions of algebraic groups on varieties

Suggested reading: Humphreys, section 8.1, 8.2, 8.3.

Lecture 6: Translation of functions, G-modules.

Suggested reading: Humphreys, section 8.5.

Assignment 2 (due Sep 01, 2023)

Lecture 7: Linearization of affine groups, Zariski tangent spaces.

Suggested reading: Humphreys, section 8.6, 5.1.

Lecture 8: Adjoint representation of Lie algebras, derivations of a Lie algebra, Lie algebras and tangent spaces.

Suggested reading: Humphreys, section 9.1.

Lecture 9: Defining the right convolution product of algebraic groups and proving an isomorphism between Lie algebras and the tangent spaces of their algebraic groups.

Suggested reading: Humphreys, section 9.2, 9.3.

Assignment 3 (due Sep 11, 2023)

Lecture 10: Differentiations of morphism of algebraic groups.

Suggested reading: Humphreys, section 9.4, 10.1.

Lecture 11: Differential of adjoint representations of algebraic groups. 

Suggested reading: Humphreys, section 10.2, 10.3, 10.4, 10.5. 

Lecture 12: Construction of certain Representations of the algebraic groups.

Suggested reading: Humphreys, section 11.1.

Lecture 13: (A theorem of Chevalley) Given a closed subgroup H of an affine algebraic group G, there is a rational representation (E, φ) of G and line L in E such that H is the stabilizer of line L in G. As a consequence of the result, the quotient G/H is a quasi-projective variety.

Suggested reading: Humphreys, section 11.2, 11.3.

Lecture 14: Let H be a closed normal subgroup of an affine algebraic group G, and there is a rational representation (E, φ) of G such that H = Ker(φ), and L(H) = Ker(dφ).  As a consequence of the result, the quotient G/H is an affine variety.

Suggested reading: Humphreys, section 11.4.

Lecture 15:  Studied the quotient spaces. Moreover, the universal mapping property for the geometric quotients.

Suggested reading: Humphreys, section 12.1, 12.2, 12.3

Midterm exam (Oct 06, 2023)

Lecture 16: Studied the Jordan decompositions for abstract affine algebraic groups.  More precisely, given G an (affine) algebraic group, consider (k[G], ρ) as the rational representation of G by the right translations. For g ∈ G, there exists a pair (gs, gu) of G such that (ρg)s= ρgs, and (ρg)u= ρgu, and g = gs gu = gu gs.

Suggested reading: Humphreys, section 15.1, 15.2, 15.3

Assignment 4 (due Oct 16, 2023)

Lecture 17:  Structure theorem of commutative linear algebraic groups.

Suggested reading: Humphreys, section 15.4, 15.5

Lecture 18:  Diagonalisable groups.  One of the main results is proved in the class is that any closed subgroup H of a d-group G can be written as an intersection of kernels of the characters of G.

Suggested reading: Humphreys, section 16.1

Lecture 19: Structure theorem of arbitrary d-groups. 

Suggested reading: Humphreys, section 16.2

Assignment  5 (due Oct 26, 2023)

Lecture 20: Rigidity theorem of diagonalisable groups 

Suggested reading: Humphreys, section 16.3, 16.4

Lecture 21: Let G be an unipotent subgroup of GL(V), V a nonzero finite dimensional vector space over an algebraically closed field. Then G has a fixed point in V. 

Suggested reading: Humphreys, section  17.5

Lecture 22: Lie Kolchin Theorem.

Suggested reading: Humphreys, section 17.6

Lecture 23: Characteristic zero (Centralisers and centre)

Suggested reading: Humphreys, section 13.2, 13.3, 13.4, 13.5

Lecture 24: Abstract Jordan Decomposition, Root space decomposition.

Suggested reading: Humphreys, section 14.1, 14.2

Lecture 25: Conjugacy classes

Assignment  6 (due Nov 15, 2023)

Lecture 26: Action of a semisimple element on a Unipotent group.

Suggested reading: Humphreys, section 18.3.

Lecture 27: Structure theorem of linear algebraic groups of dimension one.

Lecture 28:  (Borel fixed point theorem) If a connected solvable algebraic group G acts on a non-empty complete variety X, then G has a fixed point in X.

Suggested reading: Humphreys, section 21.1, 21.2.

Lecture 29: Conjugacy of Borel subgroups and maximal tori.

Suggested reading: Humphreys, section 21.3.

Lecture 30: Density Theorem 

Suggested reading: Humphreys, section 22.1, 22.2

Lecture 31: Connectedness Theorem 

Suggested reading: Humphreys, section 22.3, 22.4, 22.5.

End-sem exam (Nov 30, 2023)

Assessment: Assignments - 30%, Mid Sem - 30%, End Sem - 40%