Non-commutative algebraic structures are at the core of recent developments of fundamental mathematics, from quantum groups (Hopf algebras) to non-commutative geometry. In this project of Research School complementary aspects are considered inside this direction: homology and cohomology, Frobenius algebras, representation theory and path algebras as well as Leavitt path algebras. Those subjects as well as their articulations will provide a first strong inside to the theory.
Homology and cohomology of algebras are useful tools in various domains of mathematics; they are related to notions such as the center, the derivations, rigidity and to several fundamental conjectures for understanding associative algebraic structures. The corresponding course will enable beginners to learn from the scratch the fundamental aspects. Several examples will be considered too.
Frobenius algebras represent a wide class of algebras containing several important classes including Hopf algebras. The course will provide an elementary introduction to the theory, related to representation theory.
Representation theory is an influential bunch of mathematics, for instance it is strongly linked with particle physics. This goes hand in hand with the representation theory of Lie groups and algebras, where the notions of root system and Dynkin diagrams play a central role.
Precisely the course on root systems and representation theory will introduce basic concepts and the classification via Dynkin diagrams. It will show how Dynkin diagrams occurs in representation theory of finite dimensional associative algebras (path algebras), and how this is related to finiteness conditions on the number of indecomposable modules.
A course on path and Leavitt path algebras will also take place. After introducing Leavitt path algebras the main concern will be the Lie structure of Leavitt path algebras. The starting point will be the study of the center of a Leavitt path algebra and then the study of the derived Lie algebra modulo its center.