Pro-Lie group
In mathematics, a Pro-Lie group is a topological group that can be written as a limit of Lie groups .
The class of all Pro Lie groups contains all Lie groups , compact groups and connected locally compact groups , but is closed under any products, which often makes it easier to handle than the class of locally compact groups, for example. Locally compact Pro-Lie groups have been known since the fifth Hilbert problem was solved by Andrew Gleason , Deane Montgomery and Leo Zippin , the extension to non-locally compact Pro-Lie groups is essentially based on the book The Lie-Theory of Connected Pro-Lie Groups by Karl Heinrich Hofmann and Sidney Morris , but has now also attracted many authors.
definition
A topological group is a group with a link and a neutral element provided with a topology so that both (with the product topology on ) and the inverse formation are continuous . A Lie group is a topological group on which there is also a differentiable structure , so that the multiplication and inverse formation are smooth . Such a structure is - if it exists - always unique.
A topological group is a Pro-Lie group if and only if it has one of the following equivalent properties:
- The group is the projective limit of a family of Lie groups, taken in the category of topological groups.
- The group is topologically isomorphic to a closed subgroup of a (possibly infinite) product of Lie groups.
- The group is complete (with regard to its uniform structure on the left ) and every open environment of the one element of the group contains a closed normal divisor , so that the quotient group is a Lie group.
Note that in this article - as well as in the literature on Pro-Lie groups - a Lie group is always finite-dimensional and Hausdorff-like , but does not have to be two-countable . In particular, according to this terminology , uncountable discrete groups are ( zero-dimensional ) Lie groups and thus in particular Pro-Lie groups.
Examples
- Each Lie group is a Pro Lie group.
- With the discrete topology, each finite group becomes a (zero-dimensional) Lie group and thus in particular a Pro-Lie group.
- Every pro- finite group is thus a pro-lie group.
- Every compact group can be embedded in a product of (finite-dimensional) unitary groups and is thus a Pro-Lie group.
- Each locally compact group has an open subgroup which is a Pro-Lie group; in particular, each contiguous locally compact group is a Pro-Lie group.
- Every Abelian locally compact group is a Pro-Lie group.
- The Butcher group from numerics is a Pro-Lie group that is not locally compact.
- More generally, every character group of a (real or complex) Hopf algebra is a Pro-Lie group, which in many interesting cases is not locally compact.
- The set of all real-valued functions of a set is with the pointwise addition and the topology of pointwise convergence (product topology) an Abelian Pro-Lie group, which is not locally compact for infinite .
- The projective special linear group above the field of -adic numbers is an example of a locally compact group that is not a Pro-Lie group. This is because it is simple and thus the third condition mentioned above cannot be met in the definition of a Pro Lie group.
literature
- Karl H. Hofmann , Sidney Morris : The Lie-Theory of Connected Pro-Lie Groups . European Mathematical Society (EMS), Zurich, ISBN 978-3-03719-032-6 .