In algebraic K-theory, a field of mathematics, the Steinberg group of a ring A is the universal central extension of the commutator subgroup of the stable general linear group of A. It is named after Robert Steinberg, and it is connected with lower K-groups, notably K_{2} and K_{3}.

Definition

Abstractly, given a ring A, 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).

Presentation using generators and relations

A concrete presentation using generators and relations is as follows. Elementary matrices — i.e. matrices of the form, where \mathbf{1} is the identity matrix, is the matrix with \lambda in the (p,q)-entry and zeros elsewhere, and p \neq q — satisfy the following relations, called the Steinberg relations: The unstable Steinberg group of order r over A, 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. Mapping yields a group homomorphism. As the elementary matrices generate the commutator subgroup, this mapping is surjective onto the commutator subgroup.

Interpretation as a fundamental group

The Steinberg group is the fundamental group of the Volodin space, which is the union of classifying spaces of the unipotent subgroups of.

Relation to K-theory

K1

{K_{1}}(A) is the cokernel of the map, as K_{1} is the abelianization of and the mapping \varphi is surjective onto the commutator subgroup.

K2

{K_{2}}(A) is the center of the Steinberg group. This was Milnor's definition, and it also follows from more general definitions of higher K-groups. It is also the kernel of the mapping. Indeed, there is an exact sequence Equivalently, it is the Schur multiplier of the group of elementary matrices, so it is also a homology group:.

K3

showed that.