A Criterion for a Lie Group to Admit a Continuous Embedding in a Motion Group, A.I.Shternстатья
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Аннотация:It is proved that a connected Lie group $G$ admits a continuous embedding in a motion Liegroup (a Lie extension of an Abelian connected Lie group by a compact connectedLie 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 topologicalgroup~$L$ and an Abelian group~$A$ and this product $L\times A$ admits acompactification $Q$ such that the following diagram of Lie groups with exactrows, continuous arrows, embeddings $f_{L\times A}$ and $f_G$ and the identitymapping $f_N$ is commutative:$$\CD 0@>>>N@>\iota>>G@>\pi>>L\times A@>>>0\\ @VVV @Vf_NVV @Vf_GVV @Vf_{L\times A}VV @VVV\\ 0@>>>N@>\widetilde{\iota}>>G'@>\widetilde{\pi}>>Q@>>>0.\endCD$$Here $f_G$ is an embedding of $G$ in the motion group $G'$.