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≥3, define Stn(R) to be the free abelian group on symbols xij(r) for i,j distinct integers between 1 and n, and r∈R, subject to the following relations:
xij(r)xij(s)=xij(r+s) |
[xij,xkl]={1if j≠k and i≠lxil(rs)if j=k and i≠lxkj(-sr)if j≠k and i=l. |
Note that if eij(r) denotes the elementary matrix with one along the diagonal, and r in the (i,j) entry, then the eij(r) also satisfy the above relations, giving a well defined morphism Stn(R)→En(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)→E(R) is the second algebraic K-group of the ring R, K2(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 |
---|---|
Canonical name | SteinbergGroup |
Date of creation | 2013-03-22 16:44:38 |
Last modified on | 2013-03-22 16:44:38 |
Owner | dublisk (96) |
Last modified by | dublisk (96) |
Numerical id | 4 |
Author | dublisk (96) |
Entry type | Definition |
Classification | msc 19C09 |