Módulos e Representações

Objectives

After taking the course the students should be familiar with the basic concepts and results regarding modules and representations of algebras, as well as concrete examples of representations of finite groups and Lie algebras. They should also learn elementary notions of category theory that allow a more abstract presentation than that of introductory algebra courses.

Syllabus

Categories: products and coproducts, equalizers and co-equalizers; functors, concrete categories and free objects; isomorphism of objects that share a universal property. Modules: general definitions and examples. Linear algebra over commutative rings and integral domains: representation of homomorphisms of free modules by matrices; determinants and inverses of matrices. Smith normal form. Classification of finitely generated modules over principal ideal domains. Classification of finitely generated abelian groups. Rational and Jordan normal forms. Bimodules, tensor product, Hom and duality. Exact sequences, short-five lemma, projective, injective and flat modules. Simple and semisimple modules. Semisimple rings. Algebras. Modules over algebras. Artin-Wedderburn theorem for semisimple algebras. Representations of quivers, finite groups and Lie algebras, including sl2(C) as an example. Character tables. Orthogonality relations and consequences.

Prerequisites

This course is aimed at students who already have some knowledge about rings, but it is also accessible to students who only had a first course in linear algebra.

Cross Competence Component

The UC allows the development of transversal competences on Critical Thinking, Creativity and Problem Solving Strategies, in class, in autonomous work and in the several evaluation components. The percentage of the final grade associated with these competences should be around 15%.

Ethical Principles

All members of a group are responsible for the group’s work In any assessment, every student shall honestly disclose any help received and sources used. In an oral assessment, every student shall be able to present and answer questions about the entire assignment and solution.