Lie Algebras Research

Areas of research interest of David Towers

General nonassociative algebras

My main research interests lie in the area of algebras, where I have worked on general nonassociative algebras, Lie, Jordan and Genetic algebras. My early work concerned the Frattini subalgebra of a nonassociative algebra. This has proved to be a useful concept for studying how algebras are generated, in yielding certain decomposition theorems, and in gaining information about other subalgebra questions. My best known result from this period asserts that every subspace of a nonassociative algebra, over a field of characteristic zero, which is invariant under all automorphisms of the algebra is also invariant under all derivations of the algebra. (This has been called the Chevalley-Tuck-Towers Theorem).

Lie algebras

I have been especially active in investigating the relationship between the structure of a Lie algebra and that of its lattice of subalgebras, and in studying certain special subalgebras. A number of interesting classes of Lie algebras have been uncovered by imposing particular conditions upon certain subalgebras; in particular, new simple Lie algebras were found in this way. Of fundamental importance is the extent to which the structure of a Lie algebra is determined by lattice-theoretic properties of its subalgebras. Can important classes of algebras, such as simple, solvable and nilpotent algebras, be characterised by their subalgebra lattice? Apart from low dimensional exceptional cases, the answer is affirmative. The question of the extent to which ideals can be defined by lattice conditions leads to the study of modularity, and here I have made significant progress in collaboration with Varea and Bowman.

Genetic algebras

In the area of Genetic algebras I made progress in determining the structure of Bernstein algebras of degree k. These algebras arise in connection with the problem of characterising populations that stabilise after k generations. Algebras of degree I had been studied extensively, but little was previously known for more general degree.