Representation and Gradings of Solvable Lie Algebras

This proposal is situated in the intersection of Lie theory, representation theory and group theory. It is partially motivated by results on affine crystallographic groups and the structure of finite groups and finite-dimensional algebras. Hilbert`s Problem 18 on Euclidean space groups led to the algebraic characterisation of the crystallographic groups. This problem was then generalised to the more difficult so-called affine crystallographic groups. In this context, J. Milnor`s conjecture on virtually-polycyclic groups and affinely-flat manifolds was disproved by finding explicit counterexamples on the level of Lie algebras. These Lie algebras do not admit faithful representations that are ``small`` and it is known that counterexamples to the conjecture correspond with Lie algebras that do not admit regular one-cocycles of given dimension. Studying the consequences of the existence of regular transformations can also be motivated by different results in the theory of groups and Lie rings. The proof of the nilpotency of the Frobenius-kernel by Higman and Thompson involved regular automorphisms of prime order. An elementary proof of the coclass conjectures on p-groups by Shalev and Zel`manov was based on results on periodic derivations and automorphisms of Lie rings. Several other regularity results have been obtained from Jacobson`s theorem on weakly closed sets of nilpotent operators. My goal is to gain a better understanding of the general problems in this field and to contribute to the progress that has already been made. I think the following three suggestions are good starting points. First, I want to approximate the solvable radical of a Lie algebra with characteristic ideals of low nilpotency class in order to construct faithful representations whose degree is bounded by a polynomial in natural invariants of the Lie algebra. This will use embedding theorems, split extensions, universal enveloping algebras and quotient-by-invariants constructions. Secondly, I intend to extend results on periodic derivations by Kostrikin-Kuznetsov and Burde-Moens by considering generalised Engel conditions and group determinants. Finally, I would like to refine the general bounds on the nilpotency class (and other invariants, in characteristic zero) of finite-dimensional Lie algebras with a regular derivation.

Mathematical objects often appear more difficult or more intimidating than they really are. It is therefore useful to develop tools to find out whether a given object X is small or easy or simple in some precise sense, depending on the context. We could try to show, for ex- ample, that X is finite, or that X can be written down using small matrices, or that we can recover much information about X by looking at small pieces of X at a time. The most obvious way to obtain such a simplification is by approximating X with easier objects L1, L2, . . ..The project did precisely this.(a.) We first looked at an old conjecture in geometry that deals with repeating patterns in both art and chemistry: the so-called wallpaper groups and crystallographic groups. The conjecture predicted that X could be approximated with a small object, L, but some time ago it was shown that such an L does not always exist. In this context, we say that L is small whenever it can be written down using small matrices. In this project we proved that if L has good symmetries, then L can always be written down with reasonably small matrices.(b.) The above approximations L are known as Lie rings and Lie algebras. Again, we could show that if L has good symmetries, then L is small in other ways. In one problem, we could show that L must be finite. In another problem, we showed that if we multiply sufficiently many elements of L, then the result is always zero. In yet another problem, we could show that if L admits many good symmetries, then L itself can be approximated with an even easier object, N. In each of these problems, the information about L is then used to prove that the original object X is small as well.One of the main results of the project is a method that can be used to solve several of the classical problems at once. This is good because it shows that many old problems are, in fact, closely connected. And it is good, because it suggests that many more open problems could one day be solved by using similar techniques.