A note on commutators in the group of infinite triangular matrices over a ring

作者：Bier Agnieszka^*; Holubowski Waldemar

来源：Linear and Multilinear Algebra, 2015, 63(11): 2301-2310.

DOI：10.1080/03081087.2014.1003529

• infinite triangular matrix
• derived subgroup
• commutator width
• Vershik-Kerov group
• 20H25
• 20F12
• 20F14
• 15A23

We investigate the commutators of elements of the group UT(infinity, R) of infinite unitriangular matrices over an associative ring R with 1 and a commutative group U(R) of invertible elements. We prove that every unitriangular matrix of a specified form is a commutator of two other unitriangular matrices. As a direct consequence, we give a complete characterization of the lower central series of the group UT(infinity, R) including the width of its terms with respect to basic commutators and Engel words. With an additional restriction on the ring R, we show that the derived subgroup of T(infinity, R) coincides with the group UT(infinity, R). These results generalize the results obtained for triangular groups over a field.

• 出版日期2015-11-2