- Title
- Sophus Lie's third fundamental theorem and the adjoint functor theorem
- Creator
- Hofmann, Karl; Morris, Sidney
- Date
- 2005
- Type
- Text; Journal article
- Identifier
- http://researchonline.federation.edu.au/vital/access/HandleResolver/1959.17/57526
- Identifier
- vital:409
- Identifier
-
https://doi.org/10.1515/jgth.2005.8.1.115
- Identifier
- ISSN:1433-5883
- Abstract
- 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.; C1
- Publisher
- Walter de Gruyter
- Relation
- Journal of Group Theory Vol. 8, no. 1 (2005), p. 115-133
- Rights
- Copyright Walter de Gruyter
- Rights
- This metadata is freely available under a CCO license
- Subject
- 0101 Pure Mathematics; Mathematics; Lie group; Lie algebra; Exponential function; Topological groups
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