A structure theorem for the elementary unimodular vector group

Abstract

Given a pair of vectors v , w ∈ R r + 1 v,w\in R^{r+1} with ⟨ v , w ⟩ = v ⋅ w T = 1 \langle v,w\rangle =v\cdot w^T=1 , A. Suslin constructed a matrix S r ( v , w ) ∈ S l 2 r ( R ) S_r(v,w)\in Sl_{2^r}(R) . We study the subgroup S U m r ( R ) SUm_r(R) generated by these matrices, and its (elementary) subgroup E U m r ( R ) EUm_r(R) generated by the matrices S r ( e 1 ε , e 1 ε T − 1 ) S_r(e_1\varepsilon ,e_1\varepsilon ^{T^{-1}}) , for ε ∈ E r + 1 ( R ) \varepsilon \in E_{r+1}(R) . The basic calculus for E U m r ( R ) EUm_r(R) is developed via a key lemma, and a fundamental property of Suslin matrices is derived.