On Bitopological Lie Groups, Some Properties and Examples

We introduce a bitopological version of a Lie group by equipping a smooth Lie group with two possibly distinct compatible group topologies. The purpose is to separate the smooth structure from the global topological structures carried by the underlying group. We define bitopological Lie groups and their morphisms, investigate translations, identity components, products, subgroups and quotient structures, and give several examples. The classical theory of Lie groups is recovered when the two topologies coincide with the topology induced by the smooth structure. We also discuss some directions for further development of bitopological Lie theory.