Steinberg group

In mathematics, Steinberg group means either of two distinct, though related, constructions of the mathematician Robert Steinberg:

Group of Lie type

In mathematics, a group of Lie type is a group closely related to the group, G(k), of rational points of a reductive linear algebraic group, G, with values in the field, k. Finite groups of Lie type give the bulk of non-abelian finite simple groups. Special cases include the classical groups, the Chevalley groups, the Steinberg groups, and the Suzuki–Ree groups.

All Lie groups are groups of Lie type, but not vice-versa.

Dieudonné (1971) and Carter (1989) are standard references for groups of Lie type.

Classical groups

An initial approach to this question was the definition and detailed study of the so-called classical groups over finite and other fields by Jordan (1870). These groups were studied by L. E. Dickson and Jean Dieudonné. Emil Artin investigated the orders of such groups, with a view to classifying cases of coincidence.

A classical group is, roughly speaking, a special linear, orthogonal, symplectic, or unitary group. There are several minor variations of these, given by taking derived subgroups or central quotients, the latter yielding projective linear groups. They can be constructed over finite fields (or any other field) in much the same way that they are constructed over the real numbers. They correspond to the series A_n, B_n, C_n, D_n,²A_n, ²D_n of Chevalley and Steinberg groups.

Steinberg group (K-theory)

In algebraic K-theory, a field of mathematics, the Steinberg group of a ring is the universal central extension of the commutator subgroup of the stable general linear group of .

It is named after Robert Steinberg, and it is connected with lower -groups, notably and .

Definition

Abstractly, given a ring , the Steinberg group is the universal central extension of the commutator subgroup of the stable general linear group (the commutator subgroup is perfect and so has a universal central extension).

Concretely, it can be described using generators and relations.

Steinberg Relations

Elementary matrices — i.e. matrices of the form , where is the identity matrix, is the matrix with in the -entry and zeros elsewhere, and — satisfy the following relations, called the Steinberg relations:

The unstable Steinberg group of order over , denoted by , is defined by the generators , where and , these generators being subject to the Steinberg relations. The stable Steinberg group, denoted by , is the direct limit of the system . It can also be thought of as the Steinberg group of infinite order.