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Topological definition

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The statement of the theorem

A Lie group can be defined as a (Hausdorff) topological group that, near the identity element, looks like a transformation group, with no reference to differentiable manifolds nor topological manifolds. Precisely, a Lie group is defined as a topological group that (1) is locally isomorphic near the identities to a matrix Lie group, a closed subgroup of GL⁡(n,C)\operatorname {GL} (n,\mathbb {C} ) and (2) has at most countably many connected components. Showing the topological definition is equivalent to the usual one is technical (and the beginning readers should skip the following) but is done roughly as follows: - Given a Lie group G in the usual manifold sense, the Lie group–Lie algebra correspondence (or a version of Lie's third theorem) constructs a closed Lie subgroup G′⊂GL⁡(n,C)G'\subset \operatorname {GL} (n,\mathbb {C} ) such that G,G′G,G' share the same Lie algebra; thus, they are locally isomorphic. Hence, GG satisfies the above topological definition. - Conversely, let GG be a topological group that is a Lie group in the above topological sense and choose a matrix Lie group G′G' that is locally isomorphic to ⁠ GG ⁠ around the respective identities. Then, by a version of the closed subgroup theorem, G′G' is a real-analytic manifold and then, through the local isomorphism, G acquires a structure of a manifold near the identity element. One then shows that the group law on G can be given by formal power series; so the group operations are real-analytic and GG itself is a real-analytic manifold. The topological definition implies the statement that if two Lie groups are isomorphic as topological groups, then they are isomorphic as Lie groups. In fact, it states the general principle that, to a large extent, the topology of a Lie group together with the group law determines the geometry of the group. - ^Kobayashi & Oshima 2005, Definition 5.3 - ^Bruhat, F. (1958). "Lectures on Lie Groups and Representations of Locally Compact Groups"(PDF). Tata Institute of Fundamental Research, Bombay. Cite error: There are tags or {{efn}} templates on this page, but the references will not show without a {{reflist|group=lower-alpha}} template or {{notelist}} template (see the help page).