# Steinberg group (K-theory)

In algebraic K-theory, a field of mathematics, the Steinberg group $\operatorname {St} (A)$ 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 $\operatorname {St} (A)$  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 ${e_{pq}}(\lambda ):=\mathbf {1} +{a_{pq}}(\lambda )$ , where $\mathbf {1}$  is the identity matrix, ${a_{pq}}(\lambda )$  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:

{\begin{aligned}e_{ij}(\lambda )e_{ij}(\mu )&=e_{ij}(\lambda +\mu );&&\\\left[e_{ij}(\lambda ),e_{jk}(\mu )\right]&=e_{ik}(\lambda \mu ),&&{\text{for }}i\neq k;\\\left[e_{ij}(\lambda ),e_{kl}(\mu )\right]&=\mathbf {1} ,&&{\text{for }}i\neq l{\text{ and }}j\neq k.\end{aligned}}

The unstable Steinberg group of order $r$  over $A$ , denoted by ${\operatorname {St} _{r}}(A)$ , is defined by the generators ${x_{ij}}(\lambda )$ , where $1\leq i\neq j\leq r$  and $\lambda \in A$ , these generators being subject to the Steinberg relations. The stable Steinberg group, denoted by $\operatorname {St} (A)$ , is the direct limit of the system ${\operatorname {St} _{r}}(A)\to {\operatorname {St} _{r+1}}(A)$ . It can also be thought of as the Steinberg group of infinite order.

Mapping ${x_{ij}}(\lambda )\mapsto {e_{ij}}(\lambda )$  yields a group homomorphism $\varphi :\operatorname {St} (A)\to {\operatorname {GL} _{\infty }}(A)$ . 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 GL(A).

## Relation to K-theory

### K1

${K_{1}}(A)$  is the cokernel of the map $\varphi :\operatorname {St} (A)\to {\operatorname {GL} _{\infty }}(A)$ , as $K_{1}$  is the abelianization of ${\operatorname {GL} _{\infty }}(A)$  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 $\varphi :\operatorname {St} (A)\to {\operatorname {GL} _{\infty }}(A)$ . Indeed, there is an exact sequence

$1\to {K_{2}}(A)\to \operatorname {St} (A)\to {\operatorname {GL} _{\infty }}(A)\to {K_{1}}(A)\to 1.$

Equivalently, it is the Schur multiplier of the group of elementary matrices, so it is also a homology group: ${K_{2}}(A)={H_{2}}(E(A);\mathbb {Z} )$ .

### K3

Gersten (1973) showed that ${K_{3}}(A)={H_{3}}(\operatorname {St} (A);\mathbb {Z} )$ .