Structure of Group Rings and the Group of Units of Integral Group Rings - Lecture 1

Explore the fundamental concepts of group rings and their unit groups in this comprehensive lecture by Eric Jespers. Delve into the structure of group rings, examining their definition, properties, and relationship to ring theory. Investigate idempotents and their significance in group ring theory. Analyze the invertible elements of group rings and understand the equivalence of related problems. Learn about the augmentation mapping and its relevance to unit groups. Discover various constructions of units, including unipotent units and Bass units. Examine the connection between group rings and number theory, exploring cyclotomic units and related theorems. Gain insights into the applications of group rings in representation theory and their importance in modern algebra.

Structure of groups rings and the group of units of integral group rings
Group Ring
What is Group Ring?
Ring theory
Idempotent
What is the idea of a group ring originally?
How does it relate to this as groups isomorphic?
What are the invertible elements?
Lemma
Why these problems are equivalent?
Augmentation mapping
Why the unit group is really relevant?
Does that exist a torsion free normal subgroup say N, of UnZG ?
Black box
Via braces one gets closer and closer to this and braces are solutions of young Baxter equation
Why to look at unit groups?
Constructions of units
Unipotent unit
Construct potent element from idempotent
Notation
Bass units
Number theory
Cyclotomic units
Theorem