A certain decomposition of infinite invertible matrices over division algebras
Mai Hoang Bien, Truong Huu Dung, Nguyen Thi Thai Ha
Abstract
Let D be an infinite division ring with centre F and T(∞,D) the group of infinite upper triangular invertible matrices indexed by N×N over D. In this paper, we first show that if D is finite dimensional over F, then every element in T(∞,D) whose diagonal entries are commutators of D∖{0} is a commutator of an infinite diagonal matrix and another infinite upper triangular matrix. Some applications are also shown. For example, if dimFD≤4, then every element in the commutator subgroup [GLVK(∞,D),GLVK(∞,D)] of the Vershik-Kerov group GLVK(∞,D) is a commutator.
Topics & Concepts
Invertible matrixCommutatorMathematicsTriangular matrixDiagonalCombinatoricsGroup (periodic table)Element (criminal law)Commutator subgroupMatrix (chemical analysis)Diagonal matrixPure mathematicsDivision (mathematics)Division ringAlgebra over a fieldNormal subgroupArithmeticPhysicsGeometryComposite materialLawPolitical scienceMaterials scienceQuantum mechanicsLie conformal algebraAdvanced Topics in AlgebraFinite Group Theory ResearchAlgebraic structures and combinatorial models