Steinberg groups for Jordan pairs

This book aims to develop the theory of elementary and Steinberg groups over rings on the basis of two new foundations: an algebraic one, the theory of Jordan pairs, and a combinatorial one, the theory of 3-graded locally finite root systems. As a prerequisite of independent interest we present a general theory of groups with root groups and commutator relations, and introduce the notions of Steinberg groups and Weyl elements for such groups.

Traditionally, the study of the classical elementary groups of matrices is done, roughly speaking, in four cases: the linear, symplectic, unitary, and orthogonal case. It was A. Bak who took a great unifying step by introducing the concept of form ring which made it possible to treat these cases in a uniform manner. We hope that our work will be seen as a second unifying step.

Let us indicate the three basic ingredients of the book in more detail. The first is the theory of groups with commutator relations and their Steinberg groups. This does not require Jordan theory and can be read independently from the rest of the text. We study groups G generated by a family of subgroups, indexed by a generalized root system and satisfying certain commutator relations. Generalizing Tits' work on Steinberg groups in the setting of Kac-Moody groups, we introduce the notion of covering morphism, formally similar to coverings in algebraic topology, and show that G admits an essentially unique universal covering.

Next are the locally finite root systems, defined like the finite ones except that finiteness is replaced by local finiteness: the intersection of the root system with every finite-dimensional subspace of the ambient space is finite. Their classification is known and presents no surprises: besides the finite root systems, there are natural, infinite analogues of the five classical root systems. By using locally finite root systems we are able to bypass the traditional way of dealing with the infinite elementary and Steinberg groups: instead of first considering the finite case and then obtaining the infinite case by a limiting process, we deal with both cases at the same time.

The main novel aspect of our approach to elementary and Steinberg groups is the use of Jordan pairs, which not only introduces new techniques but also allows us to describe the groups in a uniform manner by relatively few relations. This is particularly noticeable when one compares the relations defining the unitary Steinberg groups in the work of Bak or Hahn with our description. With every Jordan pair, we associate its projective elementary group, a subgroup of the automorphism group of its Tits-Kantor-Koecher Lie algebra.

Jordan pairs and root systems are tied together by the notion of a root grading of a Jordan pair. For such objects, we define a Steinberg group and show, as our main result, under suitable assumptions on the Jordan pair and the root system, that this group is the universal central extension of the projective elementary group.

Ottmar Loos < >

Erhard Neher < >