GAP (SLA) - Chapter 1: Introduction

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1 Introduction

### 1 Introduction

This package is a collection of functions that I wrote for various research projects (e.g., [Gra08], [dG11], [GE09], [Gra11], [dGVY12]). The reason to collect them in a package is to avoid them getting lost. Secondly, I believe that the functions may be of wider interest.

Apart from this one, this manual has five chapters. The second describes various functions that have to do with root systems and Weyl groups. The third chapter describes some functions for semisimple Lie algebras and their modules. The remaining chapters all deal with a particular area.

The fourth chapter contains (descriptions of) functions for computing with the classification of the nilpotent orbits in simple Lie algebras. There are functions for creating the orbits and for computing representatives. We refer to [CM93] for an overview of the theory of nilpotent orbits in simple Lie algebras.

The fifth chapter is dedicated to finite order automorphisms of the simple Lie algebras and the corresponding θ-groups. The finite order automorphisms have been classified by Kac, up to conjugacy in the automorphism group. For the background on this we refer to [Hel78]. The classification is described in terms of so-called Kac diagrams. The package contains a function for creating all automorphisms of a given simple Lie algebra, of a given finite order.

The eigenspaces of an automorphism of finite order of a simple Lie algebra form a grading of that Lie algebra. Moreover, the 0-component is a reductive subalgebra, acting on the 1-component. The 0-component corresponds to a reductive reductive group, also acting on the 1-component. This group (with its action) is called a θ-group. It was introduced and studied in the 70-s by Vinberg ([Vin75] , [Vin76], [Vin79]) The package has a function for listing the nilpotent orbits of this group.

The last chapter has functions for working with semisimple subalgebras of semisimple Lie algebras. The package contains a database of semisimple subalgebras of the simple subalgebras of ranks up to 8. Moreover, there are functions for computing the semisimple subalgebras of semisimple Lie algebras on the fly. Finally, there are some functions for computing branching rules.

We remark that the package needs the package QuaGroup.

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