Subgroups of unitriangular groups of infinite matrices
Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 14, Tome 338 (2006), pp. 137-154
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We show, that for any associative ring $R$, the subgroup $\mathrm{UT}_r(\infty,R)$ of row finite matrices in $\mathrm{UT}(\infty,R)$, the group of all infinite dimensional (indexed by $\mathbb N$) upper unitriangular matrices over $R$, is generated by strings (block-diagonal matrices with finite blocks along the main diagonal). This allows us to define a large family of subgroups of $\mathrm{UT}_r(\infty,R)$ associated to some growth functions. The
smallest subgroup in this family, called the group of banded matrices, is generated by 1-banded simultaneous elementary transvections (a slight generalization of the usual notion of elementary transvections). We introduce a notion of net subgroups and characterize the normal net subgroups of $\mathrm{UT}(\infty,R)$.
@article{ZNSL_2006_338_a4,
author = {W. Holubowski},
title = {Subgroups of unitriangular groups of infinite matrices},
journal = {Zapiski Nauchnykh Seminarov POMI},
pages = {137--154},
publisher = {mathdoc},
volume = {338},
year = {2006},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/ZNSL_2006_338_a4/}
}
W. Holubowski. Subgroups of unitriangular groups of infinite matrices. Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 14, Tome 338 (2006), pp. 137-154. http://geodesic.mathdoc.fr/item/ZNSL_2006_338_a4/