Algebraic Conjugacy of Linear Flows on Connected Lie Groups

Abstract

We study algebraic conjugacy of linear flows on connected Lie groups. To each linear flow we associate a derivation of the Lie algebra, and we analyze how the conjugacy problem can be transferred to the infinitesimal level. In the connected and simply connected case, algebraic conjugacy is characterized by the conjugacy of the associated derivations. For non-simply connected Lie groups, we show that this condition must be supplemented by global restrictions arising from the universal covering group. More precisely, only projectable derivations define linear flows on the quotient, and only admissible Lie algebra isomorphisms descend to conjugacies between the quotient groups. This provides a framework for the algebraic classification of linear flows by computing projectable derivations and admissible automorphisms. We also describe fixed point subgroups and kernels of derivations as obstructions to algebraic conjugacy. Finally, we apply the general results to the three-dimensional Heisenberg group and to its central quotients, showing explicitly how the quotient topology restricts both the admissible derivations and the conjugating automorphisms.