Notes on simple groups of Lie type

In this previous post I recorded some (very standard) material on the structural theory of finite-dimensional complex Lie algebras (or Lie algebras for short), with a particular focus on those Lie algebras which were semisimple or simple. Among other things, these notes discussed the Weyl complete reducibility theorem (asserting that semisimple Lie algebras are the direct sum of simple Lie algebras) and the classification of simple Lie algebras (with all such Lie algebras being (up to isomorphism) of the form , , , , , , , , or ).

Among other things, the structural theory of Lie algebras can then be used to build analogous structures in nearby areas of mathematics, such as Lie groups and Lie algebras over more general fields than the complex field (leading in particular to the notion of a Chevalley group), as well as finite simple groups of Lie type, which form the bulk of the classification of finite simple groups (with the exception of the alternating groups and a finite number of sporadic groups).

In the case of complex Lie groups, it turns out that every simple Lie algebra is associated with a finite number of connected complex Lie groups, ranging from a “minimal” Lie group (the adjoint form of the Lie group) to a “maximal” Lie group (the simply connected form of the Lie group) that finitely covers , and occasionally also a number of intermediate forms which finitely cover , but are in turn finitely covered by . For instance, is associated with the projective special linear group as its adjoint form and the special linear group as its simply connected form, and intermediate groups can be created by quotienting out by some subgroup of its centre (which is isomorphic to the roots of unity). The minimal form is simple in the group-theoretic sense of having no normal subgroups, but the other forms of the Lie group are merely quasisimple, although traditionally all of the forms of a Lie group associated to a simple Lie algebra are known as simple Lie groups.

Thanks to the work of Chevalley, a very similar story holds for algebraic groups over arbitrary fields ; given any Dynkin diagram, one can define a simple Lie algebra with that diagram over that field, and also one can find a finite number of connected algebraic groups over (known as Chevalley groups) with that Lie algebra, ranging from an adjoint form to a universal form , with every form having an isogeny (the analogue of a finite cover for algebraic groups) to the adjoint form, and in turn receiving an isogeny from the universal form. Thus, for instance, one could construct the universal form of the algebraic group over a finite field of finite order.

When one restricts the Chevalley group construction to adjoint forms over a finite field (e.g. ), one usually obtains a finite simple group (with a finite number of exceptions when the rank and the field are very small, and in some cases one also has to pass to a bounded index subgroup, such as the derived group, first). One could also use other forms than the adjoint form, but one then recovers the same finite simple group as before if one quotients out by the centre. This construction was then extended by Steinberg, Suzuki, and Ree by taking a Chevalley group over a finite field and then restricting to the fixed points of a certain automorphism of that group; after some additional minor modifications such as passing to a bounded index subgroup or quotienting out a bounded centre, this gives some additional finite simple groups of Lie type, including classical examples such as the projective special unitary groups , as well as some more exotic examples such as the Suzuki groups or the Ree groups.

While I learned most of the classical structural theory of Lie algebras back when I was an undergraduate, and have interacted with Lie groups in many ways in the past (most recently in connection with Hilbert’s fifth problem, as discussed in this previous series of lectures), I have only recently had the need to understand more precisely the concepts of a Chevalley group and of a finite simple group of Lie type, as well as better understand the structural theory of simple complex Lie groups. As such, I am recording some notes here regarding these concepts, mainly for my own benefit, but perhaps they will also be of use to some other readers. The material here is standard, and was drawn from a number of sources, but primarily from Carter, Gorenstein-Lyons-Solomon, and Fulton-Harris, as well as the lecture notes on Chevalley groups by my colleague Robert Steinberg. The arrangement of material also reflects my own personal preferences; in particular, I tend to favour complex-variable or Riemannian geometry methods over algebraic ones, and this influenced a number of choices I had to make regarding how to prove certain key facts. The notes below are far from a comprehensive or fully detailed discussion of these topics, and I would refer interested readers to the references above for a properly thorough treatment.