# Steinberg group

Given an associative ring $R$ with identity, the Steinberg group $St(R)$ describes the minimal amount of relations between elementary matrices in $R$.

For $n\geq 3$, define $St_{n}(R)$ to be the free abelian group on symbols $x_{ij}(r)$ for $i,j$ distinct integers between $1$ and $n$, and $r\in R$, subject to the following relations:

 $x_{ij}(r)x_{ij}(s)=x_{ij}(r+s)$
 $[x_{ij},x_{kl}]=\begin{cases}1&\text{if j\neq k and i\neq l}\\ x_{il}(rs)&\text{if j=k and i\neq l}\\ x_{kj}(-sr)&\text{if j\neq k and i=l.}\end{cases}$

Note that if $e_{ij}(r)$ denotes the elementary matrix with one along the diagonal, and $r$ in the $(i,j)$ entry, then the $e_{ij}(r)$ also satisfy the above relations, giving a well defined morphism $St_{n}(R)\to E_{n}(R)$, where the latter is the group of elementary matrices.

Taking a colimit over $n$ gives the Steinberg group $St(R)$. The importance of the Steinberg group is that the kernel of the map $St(R)\to E(R)$ is the second algebraic $K$-group of the ring $R$, $K_{2}(R)$. This also coincides with the kernel of the Steinberg group. One can also show that the Steinberg group is the universal central extension of the group $E(R)$.

Title Steinberg group SteinbergGroup 2013-03-22 16:44:38 2013-03-22 16:44:38 dublisk (96) dublisk (96) 4 dublisk (96) Definition msc 19C09