A CRITERION FOR A LIE GROUP TO ADMIT A CONTINUOUS EMBEDDING IN A MOTION GROUP

A CRITERION FOR A LIE GROUP TO ADMIT A CONTINUOUS EMBEDDING IN A MOTION GROUP

It is proved that a connected Lie group G admits a continuous embedding in a motion Lie group (a Lie extension of an Abelian connected Lie group by a compact connected Lie group) if and only if the Lie group G has an Abelian normal subgroup N such that the quotient group G/N is a direct product of a compact topological group L and an Abelian group A and this product L × A admits a compactification Q such that the following diagram of Lie groups with exact rows, continuous arrows, embeddings fL×A and fG and the identity mapping fN is commutative: Here fG is an embedding of G in the motion group G".

It is proved that a connected Lie group G admits a continuous embedding in a motion Lie group (a Lie extension of an Abelian connected Lie group by a compact connected Lie group) if and only if the Lie group G has an Abelian normal subgroup N such that the quotient group G/N is a direct product of a compact topological group L and an Abelian group A and this product L × A admits a compactification Q such that the following diagram of Lie groups with exact rows, continuous arrows, embeddings fL×A and fG and the identity mapping fN is commutative: Here fG is an embedding of G in the motion group G".