# Steinberg group (K-theory)

## 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).

Concretely, it can be described using generators and relations.

### 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}} ## Relation to $K$ -Theory

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

$1\to {K_{2}}(A)\to \operatorname {St} (A)\to {\operatorname {GL} _{\infty }}(A)\to {K_{1}}(A)\to 1.$ Equivalently, it is the Schur multiplier of the group of elementary matrices, so it is also a homology group: ${K_{2}}(A)={H_{2}}(E(A);\mathbb {Z} )$ .