Steinberg group

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

For n3, define Stn(R) to be the free abelian groupMathworldPlanetmath on symbols xij(r) for i,j distinct integers between 1 and n, and rR, subject to the following relations:

[xij,xkl]={1if jk and ilxil(rs)if j=k and ilxkj(-sr)if jk 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