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Project
Groups, Group ringen and the Yang-Baxter equation (FWOAL630)
Let R be a ring and G a group. Then the group ring RG is an associative ring that is a free R-module with basis G. This ring is a fundamental tool in the link between group theory and ring theory.
The importance of group rings already became apparent in the work of Molien, Frobenius, Schur and Maschke. The study of group rings also involves several other fields, such as field theory, linear algebra, algebraic number theory, K-theory, algebraic topology, homological algebra. Of particular importance is the integral group ring ZG and its group U of invertible elements. The latter allows one, for many finite groups G, to recover the group G from the ring ZG.
In this project we investigate fundamental problems on the unit group of ZG. To do so we need to consider the unit group of the wider class of orders in finite dimensional rational simple algebras. A main theme throughout the project is to obtain a presentation of U, i.e. generators and relations. In case G is infinite and torsion free it is conjectured that U consists of trivial units only. The groups that describe non-degenerate set theoretic solutions of the Yang-Baxter equation are an ideal testing ground for this conjecture. They are of fundamental interest on their own and hence their structure will be further investigated as well.
The importance of group rings already became apparent in the work of Molien, Frobenius, Schur and Maschke. The study of group rings also involves several other fields, such as field theory, linear algebra, algebraic number theory, K-theory, algebraic topology, homological algebra. Of particular importance is the integral group ring ZG and its group U of invertible elements. The latter allows one, for many finite groups G, to recover the group G from the ring ZG.
In this project we investigate fundamental problems on the unit group of ZG. To do so we need to consider the unit group of the wider class of orders in finite dimensional rational simple algebras. A main theme throughout the project is to obtain a presentation of U, i.e. generators and relations. In case G is infinite and torsion free it is conjectured that U consists of trivial units only. The groups that describe non-degenerate set theoretic solutions of the Yang-Baxter equation are an ideal testing ground for this conjecture. They are of fundamental interest on their own and hence their structure will be further investigated as well.
Date:1 Jan 2012 → 31 Dec 2015
Keywords:Mathematics
Disciplines:Mathematical sciences and statistics