# Steinberg group (K-theory)

In algebraic K-theory, a field of mathematics, the Steinberg group ${\displaystyle \operatorname {St} (A)}$ of a ring ${\displaystyle A}$ is the universal central extension of the commutator subgroup of the stable general linear group of ${\displaystyle A}$.

It is named after Robert Steinberg, and it is connected with lower ${\displaystyle K}$-groups, notably ${\displaystyle K_{2}}$ and ${\displaystyle K_{3}}$.

## Definition

Abstractly, given a ring ${\displaystyle A}$ , the Steinberg group ${\displaystyle \operatorname {St} (A)}$  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 ${\displaystyle {e_{pq}}(\lambda ):=\mathbf {1} +{a_{pq}}(\lambda )}$ , where ${\displaystyle \mathbf {1} }$  is the identity matrix, ${\displaystyle {a_{pq}}(\lambda )}$  is the matrix with ${\displaystyle \lambda }$  in the ${\displaystyle (p,q)}$ -entry and zeros elsewhere, and ${\displaystyle p\neq q}$  — satisfy the following relations, called the Steinberg relations:

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

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

${\displaystyle {K_{1}}(A)}$  is the cokernel of the map ${\displaystyle \varphi :\operatorname {St} (A)\to {\operatorname {GL} _{\infty }}(A)}$ , as ${\displaystyle K_{1}}$  is the abelianization of ${\displaystyle {\operatorname {GL} _{\infty }}(A)}$  and the mapping ${\displaystyle \varphi }$  is surjective onto the commutator subgroup.

### K2

${\displaystyle {K_{2}}(A)}$  is the center of the Steinberg group. This was Milnor's definition, and it also follows from more general definitions of higher ${\displaystyle K}$ -groups.

It is also the kernel of the mapping ${\displaystyle \varphi :\operatorname {St} (A)\to {\operatorname {GL} _{\infty }}(A)}$ . Indeed, there is an exact sequence

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

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

### K3

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

## References

• Gersten, S. M. (1973), "${\displaystyle K_{3}}$  of a Ring is ${\displaystyle H_{3}}$  of the Steinberg Group", Proceedings of the American Mathematical Society, American Mathematical Society, 37 (2): 366–368, doi:10.2307/2039440, JSTOR 2039440