The aim of the course is to provide an introduction to commutative algebra and to the theory of Lie algebras.

Lectures

Oral Examination

The contents of the courses: Algebra 1, Algebra 2, Linear Algebra and Geometry 1.

Commutative algebra:
Modules over a (commutative) ring and operations on modules; tensor product of modules. Localization of rings and modules. Primary decomposition of ideals. Artinian and Noetherian rings and modules. Dimension theory. Integral dependence and valuations; Dedekind domains. Spectrum of a commutative ring; affine algebraic sets, Noether's normalization lemma and Hilbert's Nullstellensatz.
Lie algebras:
Semisimple endomorphisms; Jordan-Chevalley decomposition. Lie algebras and Lie groups. Ideals and subalgebras. Solvable and nilpotent Lie algebras. The theorems of Lie and Engel. Cartan subalgebras. Linear representations of Lie groups and algebras. Representations of sl(2,C). Semisimple Lie algebras. Semisimplicity criteria. Root systems and their classification. Classical Lie algebras. Exceptional Lie algebras. Finite-dimensional representations of semisimple Lie algebras.

M.F. Atiyah, I.G. MacDonald: "Introduzione all'algebra commutativa", Feltrinelli, 1981.
S. Bosch: "Algebraic Geometry and Commutative Algebra", Universitext, Springer, 2013.
I. Kaplanski: "Commutative Rings", University of Chicago Press, 1974.
H. Matsumura: "Commutative Ring Theory", Cambridge University Press, 1989.
Lie algebras:
K. Erdmann, M.J. Wildon, Introduction to Lie Algebras, Springer 2006
J. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer 1972
J.P. Serre, Algebres de Lie semi-simples complexes, Benjamin 1966
A. Kirillov, Introduction to Lie Groups and Lie Algebras, https://www.math.stonybrook.edu/~kirillov/mat552/liegroups.pdf
J. Bernstein, Lectures on Lie Algebras, http://www.math.tau.ac.il/~bernstei/Publication_list/publication_texts/bernsteinLieNotes_book.pdf