Automorphic Lie Algebras and their invariants

Sara Lombardo (Northumbria)

Tuesday 10th March, 2015 15:00-16:00 Maths 516

Abstract

The concept of Automorphic Lie Algebras (ALiAs) arises in the context of reduction groups introduced in the late 1970s in the field of integrable systems. ALiAs are obtained by imposing a finite group symmetry on a Lie algebra over a field of rational functions. Since their introduction the goal has been to classify them. Past work shows remarkable uniformity between the Lie algebras associated to different reduction groups. For example, if the base Lie algebra consists of traceless 2 × 2 matrices then the ALiA is independent of the reduction group. Properties that are independent of the reduction group are called invariants. In this talk I discuss the classification results obtained so far. They indicate that if two ALiAs can be isomorphic as Lie Algebras, based on the (co)dimension counts, they are.
After introducing the concept of invariants, I will discuss those found so far and use them to set up a structure theory for Automorphic Lie Algebras. This naturally leads to a cohomology theory for root systems, which will be presented if time allows. A first exploration of this structure theory narrows down the search for Automorphic Lie Algebras significantly. Various particular cases are fully determined by their invariants, including most of the previously studied Automorphic Lie Algebras, thereby providing an explanation for their uniformity. From a more general perspective, the success of the structure theory and root system cohomology in absence of a field promises interesting theoretical developments for Lie algebras over a ring.
The talk is based on joint work with Vincent Knibbeler (Northumbria University Newcastle) and Jan Sanders (Vrije Universiteit Amsterdam).

 

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