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Open mapping theorem for topological groups

- Hofmann, Karl, Morris, Sidney

**Authors:**Hofmann, Karl , Morris, Sidney**Date:**2007**Type:**Text , Journal article**Relation:**Topology Proceedings Vol. 31, no. 2 (2007), p. 533-551**Full Text:**false**Reviewed:****Description:**We survey sufficient conditions that force a surjective continuous homomorphism between topological groups to be open. We present the shortest proof yet of an open mapping theorem between projective limits of finite dimensional Lie groups.**Description:**C1**Description:**2003005915

Sophus Lie's third fundamental theorem and the adjoint functor theorem

- Hofmann, Karl, Morris, Sidney

**Authors:**Hofmann, Karl , Morris, Sidney**Date:**2005**Type:**Text , Journal article**Relation:**Journal of Group Theory Vol. 8, no. 1 (2005), p. 115-133**Full Text:**false**Reviewed:****Description:**The essential attributes of a Lie group G are the associated Lie algebra LðGÞ and the exponential function exp : LðGÞ ! G. The prescription L operates not only on Lie groups but also on morphisms between them: it is a functor. Many features of Lie theory are shared by classes of topological groups which are much larger than that of Lie groups; these classes include the classes of compact groups, locally compact groups, and pro-Lie groups, that is, complete topological groups having arbitrarily small normal subgroups N such that G=N is a (finitedimensional) Lie group. Considering the functor L it is therefore appropriate to contemplate more general classes of topological groups. Certain functorial properties of the assignment of a Lie algebra to a topological group (where possible) will be essential. What is new here is that we will introduce a functorial assignment from Lie algebras to groups and investigate to what extent it is inverse to the Lie algebra functor L. While the Lie algebra functor is well known and is cited regularly, the existence of a Lie group functor available to be cited and applied appears less well known. Sophus Lie’s Third Fundamental Theorem says that for each finite-dimensional real Lie algebra there is a Lie group whose Lie algebra is (isomorphic to) the given one; but even in classical circumstances it is not commonly known that this happens in a functorial fashion and what the precise relationship between the Lie algebra functor and the Lie group functor is.**Description:**C1**Description:**2003001415

The structure of abelian pro-Lie groups

- Hofmann, Karl, Morris, Sidney

**Authors:**Hofmann, Karl , Morris, Sidney**Date:**2004**Type:**Text , Journal article**Relation:**Mathematische Zeitschrift Vol. 248, no. 4 (Dec 2004), p. 867-891**Full Text:**false**Reviewed:****Description:**A pro-Lie group is a projective limit of a projective system of finite dimensional Lie groups. A prodiscrete group is a complete abelian topological group in which the open normal subgroups form a basis of the filter of identity neighborhoods. It is shown here that an abelian pro-Lie group is a product of (in general infinitely many) copies of the additive topological group of reals and of an abelian pro-Lie group of a special type; this last factor has a compact connected component, and a characteristic closed subgroup which is a union of all compact subgroups; the factor group modulo this subgroup is pro-discrete and free of nonsingleton compact subgroups. Accordingly, a connected abelian pro-Lie group is a product of a family of copies of the reals and a compact connected abelian group. A topological group is called compactly generated if it is algebraically generated by a compact subset, and a group is called almost connected if the factor group modulo its identity component is compact. It is further shown that a compactly generated abelian pro-Lie group has a characteristic almost connected locally compact subgroup which is a product of a finite number of copies of the reals and a compact abelian group such that the factor group modulo this characteristic subgroup is a compactly generated prodiscrete group without nontrivial compact subgroups.**Description:**C1**Description:**2003000910

Projective limits of finite-dimensional Lie groups

- Hofmann, Karl, Morris, Sidney

**Authors:**Hofmann, Karl , Morris, Sidney**Date:**2003**Type:**Text , Journal article**Relation:**Proceedings of the London Mathematical Society Vol. 87, no. 3 (Nov 2003), p. 647-676**Full Text:**false**Reviewed:****Description:**For a topological group G we define N to be the set of all normal subgroups modulo which G is a finite-dimensional Lie group. Call G a pro-Lie group if, firstly, G is complete, secondly, N is a filter basis, and thirdly, every identity neighborhood of G contains some member of N. It is easy to see that every pro-Lie group G is a projective limit of the projective system of all quotients of G modulo subgroups from N. The converse implication emerges as a difficult proposition, but it is shown here that any projective limit of finite-dimensional Lie groups is a pro-Lie group. It is also shown that a closed subgroup of a pro-Lie group is a pro-Lie group, and that for any closed normal subgroup N of a pro-Lie group G, for any one parameter subgroup Y : R G/N there is a one parameter subgroup X : R G such that X(t) N = Y(t) for any real number t. The category of all pro-Lie groups and continuous group homomorphisms between them is closed under the formation of all limits in the category of topological groups and the Lie algebra functor on the category of pro-Lie groups preserves all limits and quotients.**Description:**C1**Description:**2003000376

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