Almost inner derivations of Lie algebras
A classical problem in spectral geometry was to determine whether or not isospectral manifolds are necessarily isometric. It turns out that the answer to this question is negative and several counterexamples have been given. For the construction of continuous families of isospectral and non-isometric manifolds, class preserving automorphisms of nilpotent Lie groups were crucial.
An automorphism of a group is called class preserving if and only if every element is conjugate to its image. So this condition is related to, but less strict than the one for an inner automorphism. A nilpotent Lie group admitting a discrete and cocompact subgroup and a class preserving automorphism which is not inner can be used to construct a continuous family of isospectral but non-isometric nilmanifolds. Class preserving automorphisms of a Lie group are very closely related to almost inner derivations of the corresponding Lie algebra. These are derivations for which each element is mapped to the Lie bracket of itself with some other element. The set of all almost inner derivations forms a Lie subalgebra of the derivation algebra and contains the inner derivations.
Up till now, almost inner derivations of Lie algebras have not been studied in detail yet. They have almost only been considered from a differential geometric point of view, where the focus was on constructing some examples. The goal of this thesis is to study this notion in a purely algebraic way. Although the motivation from spectral geometry only makes sense for nilpotent Lie algebras, from an algebraic point of view, there is no reason to restrict to this class only. Hence, we study almost inner derivations of Lie algebras more generally. We also consider non-nilpotent Lie algebras and Lie algebras over an arbitrary field, so not only real and complex Lie algebras.
This dissertation consists of three main parts and one appendix. The first part is an introduction, which provides all preliminaries to understand the rest. In Chapter 2, we define Lie algebras and develop the necessary notions which will be important in the study of almost inner derivations. Chapter 3 contains more information about the geometric motivation from spectral theory. We also present some properties of the related notion of class preserving automorphisms of groups. Finally, in Chapter 4, we describe some interesting techniques for doing computations on the almost inner derivations of (a class of) Lie algebras.
In the second part, we will focus on the fact that the dimension of the set of almost inner derivations depends on the field over which the Lie algebra is defined. Chapter 5 contains an elaborated example, where a Lie algebra is given by means of the structure constants. The distinction for various fields has to do with a different factorisation of polynomials. In Chapter 6, we show a procedure to construct new almost inner derivations by using finite field extensions. In particular, this gives a way to set up a Lie algebra for which the dimension of the set of almost inner derivations is distinct when we consider different fields. Chapter 7 focuses on Lie algebras related to finite groups. We explain the connection with class preserving automorphisms of finite groups and compare the results we have for the two notions.
In the last part, we will use the observations from the two other parts to compute the set of almost inner derivations for different classes of Lie algebras. In Chapter 8, we give an overview of almost inner derivations for low-dimensional Lie algebras. The appendix contains tables where the non-vanishing Lie brackets for a lot of low-dimensional Lie algebras are collected. Each time, we also provide tables with results, such as the dimension of some subalgebras of the derivation algebra. The next three chapters are devoted to other classes of nilpotent Lie algebras. Two-step nilpotent Lie algebras are studied in Chapter 9. Further, we also consider filiform Lie algebras and free nilpotent Lie algebras (in Chapter 10 respectively Chapter 11). The last chapter contains results for some other classes of (not only nilpotent) Lie algebras.