双射保持算子与单三角群的自同构
Bijections preserving commutators and automorphisms of unitriangular group
摘要：
本文给出了域$F$上单三角矩阵群$UT(n,F)$中保双射算子（PC-映射）的刻画，其中$n$是自然数或无穷大.最近M.Chen，D.Wang，and H.Zhai（2011）对有限$n$和R.Slowik（2013）对$n=\infty$描述了几乎等同的PC-映射。几乎恒等映射是保持初等横截的映射。证明了几乎恒等PC-映射是一个中心元的乘法。特别地，如果$n=\infty$，则几乎恒等映射是恒等的。与R.Slowik的结果一起，这表明$UT(\infty, F)$的任何PC-映射都是自同构。
英文摘要：
We complete characterization of bijections preserving commutators (PC-maps) in the group of unitriangular matrices $UT(n,F)$ over a field $F$, where $n$ is a natural number or infinity. PC-maps were recently described up to almost identity PC-maps by M.Chen, D.Wang, and H.Zhai (2011) for finite $n$ and by R.Slowik (2013) for $n=\infty$. An almost identity map is a map, preserving elementary transvections. We show that an almost identity PC-map is a multiplication by a central element. In particular, if $n=\infty$, then an almost identity map is identity. Together with the result of R.Slowik this shows that any PC-map of $UT(\infty, F)$ is an automorphism.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
