Matrix Lie theory and measure chains

摘要

In this paper we investigate the relation between Lie matrix groups MG≔MG(n,K) of n×n-matrices with K-entries and their so-called Lie h-algebras Lh[MG]. A Lie h-algebra for a matrix group MG≔MG(n,K) is defined as the set {A∈Mat(n,K)|Ah+I∈MG} if h>0 and {A∈Mat(n,K)|exp(At)∈MG for all t∈R}, if h=0, where Mat(n,K) is the set of n×n-matrices with K-entries and I∈Mat(n,K) is the identity matrix. We investigate the structural relations between the matrix group MG and its Lie h-algebras. There are results concerning the group and topological isomorphy of the group and its Lie h-algebra, for h>0, the subsequent h→0 limiting behaviour. Hereby we can reveal the exact relation between the commutator in MG and the Lie bracket in L0[MG]. In a final result we give a condition on the linear operator function in XΔ=A(t)·X, X(τ)=I which keeps the solution trajectory in MG.