TY - JOUR

T1 - Commutators of elementary subgroups: curiouser and curiouser

AU - Vavilov, Nikolai

AU - Zhang, Zuhong

PY - 2021/6/17

Y1 - 2021/6/17

N2 - Let $R$ be any associative ring with $1$, $n\ge 3$, and let $A,B$ be two-sided ideals of $R$. In our previous joint works with Roozbeh Hazrat [17,15] we have found a generating set for the mixed commutator subgroup $[E(n,R,A),E(n,R,B)]$. Later in [29,34] we noticed that our previous results can be drastically improved and that $[E(n,R,A),E(n,R,B)]$ is generated by 1) the elementary conjugates $z_{ij}(ab,c)=t_{ij}(c)t_{ji}(ab)t_{ij}(-c)$ and $z_{ij}(ba,c)$, 2) the elementary commutators $[t_{ij}(a),t_{ji}(b)]$, where $1\le i\neq j\le n$, $a\in A$, $b\in B$, $c\in R$. Later in [33,35] we noticed that for the second type of generators, it even suffices to fix one pair of indices $(i,j)$. Here we improve the above result in yet another completely unexpected direction and prove that $[E(n,R,A),E(n,R,B)]$ is generated by the elementary commutators $[t_{ij}(a),t_{hk}(b)]$ alone, where $1\le i\neq j\le n$, $1\le h\neq k\le n$, $a\in A$, $b\in B$. This allows us to revise the technology of relative localisation, and, in particular, to give very short proofs for a number of recent results, such as the generation of partially relativised elementary groups $E(n,A)^{E(n,B)}$, %% normality of $E(n,AB+BA)$ inside $[E(n,R,A),E(n,R,B)]$, multiple commutator formulas, commutator width, and the like.

AB - Let $R$ be any associative ring with $1$, $n\ge 3$, and let $A,B$ be two-sided ideals of $R$. In our previous joint works with Roozbeh Hazrat [17,15] we have found a generating set for the mixed commutator subgroup $[E(n,R,A),E(n,R,B)]$. Later in [29,34] we noticed that our previous results can be drastically improved and that $[E(n,R,A),E(n,R,B)]$ is generated by 1) the elementary conjugates $z_{ij}(ab,c)=t_{ij}(c)t_{ji}(ab)t_{ij}(-c)$ and $z_{ij}(ba,c)$, 2) the elementary commutators $[t_{ij}(a),t_{ji}(b)]$, where $1\le i\neq j\le n$, $a\in A$, $b\in B$, $c\in R$. Later in [33,35] we noticed that for the second type of generators, it even suffices to fix one pair of indices $(i,j)$. Here we improve the above result in yet another completely unexpected direction and prove that $[E(n,R,A),E(n,R,B)]$ is generated by the elementary commutators $[t_{ij}(a),t_{hk}(b)]$ alone, where $1\le i\neq j\le n$, $1\le h\neq k\le n$, $a\in A$, $b\in B$. This allows us to revise the technology of relative localisation, and, in particular, to give very short proofs for a number of recent results, such as the generation of partially relativised elementary groups $E(n,A)^{E(n,B)}$, %% normality of $E(n,AB+BA)$ inside $[E(n,R,A),E(n,R,B)]$, multiple commutator formulas, commutator width, and the like.

KW - полная линейная группа, элементарные подгруппы, конгруэнц-подгруппы, стандартная коммутационная формула, нерелятивизованная коммутационная формула, элементарные образующие

UR - https://www.researchgate.net/publication/340963074_Commutators_of_elementary_subgroups_curiouser_and_curiouser

M3 - Article

JO - Transformation Groups

JF - Transformation Groups

SN - 1083-4362

ER -