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. ${\displaystyle G}$ ${\displaystyle \cdot }$ ${\displaystyle e}$ ${\displaystyle \cdot \colon G\times G\to G}$ ${\displaystyle G\times G}$ ${\displaystyle x\mapsto x^{-1}}$

A topological group is a Pro-Lie group if and only if it has one of the following equivalent properties: ${\displaystyle G}$

• The group is the projective limit of a family of Lie groups, taken in the category of topological groups. ${\displaystyle G}$
• The group is topologically isomorphic to a closed subgroup of a (possibly infinite) product of Lie groups. ${\displaystyle G}$
• 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. ${\displaystyle U}$ ${\displaystyle N\subseteq U}$ ${\displaystyle G/N}$

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 . ${\displaystyle \mathbb {R} ^{J}}$ ${\displaystyle J}$ ${\displaystyle J}$
• 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. ${\displaystyle \operatorname {PSL} _{2}(\mathbb {Q} _{p})}$ ${\displaystyle p}$

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 .