Lifting Automorphisms from Root Systems to Lie Algebras

Abstract

In 1996 and 2000 A.G. Helminck gave the first algorithms for computing some of the structure of symmetric spaces. In this thesis we extend these results by designing algorithms for other aspects of the structure of local symmetric spaces. We begin with an involution on the root system. We would like to understand how this involution describes an involution on the Lie algebra. To do so, we consider the concept of lifting. We say an involution θ on the root system Φ can be lifted to an involution θ on the algebra if we can find θ so that θ|Φ = θ. Success gives rise to a method to compute local symmetric spaces.

Accomplishing this task requires effort on multiple fronts. On a small scale we consider a correction vector. A correction vector lives in the toral subalgebra of the Lie algebra. A result due to Steinberg establishes a unique Lie algebra automorphism that can always be defined. We can modify this map with the correction vector so that it becomes an involution.

On a large scale, computing the correction vector is too cumbersome. We will show how to “break apart†larger involutions on the root system by projecting the roots into the local symmetric space, then “extracting†specific sub-systems. We can correct the involution on each sub-system, then “glue†the pieces together to form the involution on the whole algebra. This process not only vastly improves the timing of the lifting process, but also gives rise to an argument that any involution on the root system can be lifted.

We then present an entire computer package written for Mathematica) for working with local symmetric spaces. This package includes the algorithms we devise, as well as “helper†algorithms which are necessary for implementation.