论文标题
相对和不依赖基本统一群体的换向因子
Commutators of relative and unrelative elementary unitary groups
论文作者
论文摘要
在本文中，我们发现了相对基本组的混合换向器亚组的生成器，并在Bak的统一组的环境中获得了换向器公式的不叠加版本。它是我们类似结果的直接续集，分别是$ gl（n，r）$的，分别是1分，雪佛兰组分别具有1个。也就是说，让$（a，λ）$为任何表格环和$ n \ ge 3 $。我们认为Bak的双曲线统一组$ gu（2n，a，λ）$。此外，令$（i，γ）$为$（a，λ）$的理想形式。一个人可以与$（i，γ）$相关联，相应的基本亚组$ fu（2n，i，γ）$和$ gu（2n，a，λ）$的相对基本亚组$ eu（2n，i，γ）$。令$（j，δ）$是$（a，λ）$的另一种理想形式。在本文中，我们证明了出乎意料的结果，即$ \ big [eu（2n，i，γ），eu（2n，j，Δ）\ big] $的非明显类型的发电机类型，如我们以前与Hazrat的论文中所构建的，是冗余的，并且可以作为明显的生成器的产品表示，可以表达为$ z_ {ij}（ab，c）= t_ {ji}（c）t_ {ij}（ab）t_ {ji}（ - c）$和$ z_ {ij}（ij}（ba，c）$，以及基本交换器$ y_ y_ y_ y_ y_ {ij}（ij}（a，b）= [a，b）= [t_ ij ij ij i} $ a \ in（i，γ）$，$ b \ in（j，δ）$，$ c \ in（a，λ）$。因此，$ \ big [fu（2n，i，γ），fu（2n，j，δ）\ big] = \ big [eu（2n，i，γ），eu（2n，j，j，j，Δ）\ big] $。实际上，我们建立了更精确的生成结果。特别是，即使是基本换向器$ y_ {ij}（a，b）$也应该用于一个长根位置和一个短根位置。此外，$ y_ {ij}（a，b）$是中央模型$ eu（2n，（i，γ）\ circ（j，δ））$，并且表现为符号。这使我们能够概括和统一许多以前的结果，包括多个基本换向器公式，并大大简化其证明。
In the present paper we find generators of the mixed commutator subgroups of relative elementary groups and obtain unrelativised versions of commutator formulas in the setting of Bak's unitary groups. It is a direct sequel of our similar results were obtained for $GL(n,R)$ and for Chevalley groups over a commutative ring with 1, respectively. Namely, let $(A,Λ)$ be any form ring and $n\ge 3$. We consider Bak's hyperbolic unitary group $GU(2n,A,Λ)$. Further, let $(I,Γ)$ be a form ideal of $(A,Λ)$. One can associate with $(I,Γ)$ the corresponding elementary subgroup $FU(2n,I,Γ)$ and the relative elementary subgroup $EU(2n,I,Γ)$ of $GU(2n,A,Λ)$. Let $(J,Δ)$ be another form ideal of $(A,Λ)$. In the present paper we prove an unexpected result that the non-obvious type of generators for $\big[EU(2n,I,Γ),EU(2n,J,Δ)\big]$, as constructed in our previous papers with Hazrat, are redundant and can be expressed as products of the obvious generators, the elementary conjugates $Z_{ij}(ab,c)=T_{ji}(c)T_{ij}(ab)T_{ji}(-c)$ and $Z_{ij}(ba,c)$, and the elementary commutators $Y_{ij}(a,b)=[T_{ji}(a),T_{ij}(b)]$, where $a\in(I,Γ)$, $b\in(J,Δ)$, $c\in(A,Λ)$. It follows that $\big[FU(2n,I,Γ),FU(2n,J,Δ)\big]= \big[EU(2n,I,Γ),EU(2n,J,Δ)\big]$. In fact, we establish much more precise generation results. In particular, even the elementary commutators $Y_{ij}(a,b)$ should be taken for one long root position and one short root position. Moreover, $Y_{ij}(a,b)$ are central modulo $EU(2n,(I,Γ)\circ(J,Δ))$ and behave as symbols. This allows us to generalise and unify many previous results,including the multiple elementary commutator formula, and dramatically simplify their proofs.
