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 $j\eq k$ and $i\eq l$}\\\\x_{il}(rs)&\\text{if $j=k$ and $i\eq l$}\\\\x_{kj}(-sr)&\\text{if $j\eq k$ 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 $E(R)$ .