اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدOn small abstract quotients of Lie groups and locally compact groups
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تأليف
Linus Kramer
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We prove continuity results for abstract epimorphisms of locally compact groups onto finitely generated groups.
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We are concerned with questions of the following type. Suppose that $G$ and $K$ are topological groups belonging to a certain class $cal K$ of spaces, and suppose that $phi:K to G$ is an abstract (i.e. not necessarily continuous) surjective group hom
omorphism. Under what conditions on the group $G$ and the kernel is the homomorphism $phi$ automatically continuous and open? Questions of this type have a long history and were studied in particular for the case that $G$ and $K$ are Lie groups, compact groups, or Polish groups. We develop an axiomatic approach, which allows us to resolve the question uniformly for different classes of topological groups. In this way we are able to extend the classical results about automatic continuity to a much more general setting.
Suppose that $X=G/K$ is the quotient of a locally compact group by a closed subgroup. If $X$ is locally contractible and connected, we prove that $X$ is a manifold. If the $G$-action is faithful, then $G$ is a Lie group.
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We show that, in compact semisimple Lie groups and Lie algebras, any neighbourhood of the identity gets mapped, under the commutator map, to a neighbourhood of the identity.
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