Dense Lie group homomorphisms
This article gives a survey of the basic facts on the structure and properties of dense Lie group homomorphisms, continuous homomorphisms between connected Lie groups with dense images. Dense images of vector spaces in tori are widely known examples. The treatment leads to the description of all dense embeddings of a given Lie group into other Lie groups, thus solving a problem of Goto. Among results that the author obtains, it is shown that if \(\varphi : H \to G\) is a dense continuous homomorphism of connected Lie groups, then \(G = \varphi (H)T\) for any maximal torus \(T\) in \(G\). In particular, the different maximal tori in \(G\) are conjugate by elements in \(\varphi (H)\). The classification of dense embeddings of \(H\) is reduced to an abelian problem concerning a given dense homomorphism \(\alpha : U \to T\), where \(T\) is a maximal torus in the closure of \(\text{Ad} (H)\), the image of adjoint representation, and \(U\) is its inverse (under Ad) in \(H\).
scientific article; zbMATH DE number 682587