The classical result, due to Jordan, Burnside, Dickson, says that every normal subgroup of GLn(K) (K a field, n ≥ 3) which is not contained in the centre, contains SLn(K). A. Rosenberg gave description of normal subgroups of GL(V), where V is a vector space of any infinite cardinality dimension over a division ring. However, when he considers subgroups of the direct product of the centre and the group of linear transformations g such that g − idV has finite-dimensional range the proof is not complete. We fill this gap for countable-dimensional V giving a description of the lattice of normal subgroups in the group of infinite column-finite matrices indexed by positive integers over any field. Similar results for Lie algebras of matrices will be surveyed.
The talk is based on results presented in this first arXiv paper for groups and this second paper for Lie algebras. Joint work with Martyna Maciaszczyk and Sebastian Zurek.
This video was produced by Newcastle University, Australia, as part of the Symmetries in Newcastle seminar series.
