Draft:Pro-Lie Group

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A pro-Lie group is in mathematics a topological group that can be written in a certain sense 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 arbitrary products, which often makes it easier to handle than, for example, the class of locally compact groups. Locally compact pro-Lie groups have been known since the solution of the fifth Hilbert problem by Andrew Gleason, Deane Montgomery and Leo Zippin, the extension to nonlocally compact pro-Lie groups is essentially due to the book The Lie-Theory of Connected Pro-Lie Groups by Karl Heinrich Hofmann and Sidney Morris, but has since attracted many authors.

Definition

A topological group is a group $G$ with multiplication $\cdot$ and neutral element $e$ provided with a topology such that both $\cdot :G\times G\to G$ (with the product topology on $G\times G$) and the inverse map $x\mapsto x^{-1}$ are continuous. A Lie group is a topological group on which there is also a differentiable structure such that the multiplication and inverse are smooth. Such a structure – if it exists – is always unique.

A topological group $G$ is a pro-Lie group if and only if it has one of the following equivalent properties:

• The group $G$ is the projective limit of a family of Lie groups, taken in the category of topological groups.
• The group $G$ is topologically isomorphic to a closed subgroup of a (possibly infinite) product of Lie groups.
• The group is complete (with respect to its left uniform structure) and every open neighborhood $U$ of the unit element of the group contains a closed normal subgroup $N\subseteq U$, so that the quotient group $G/N$ 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 Hausdorffian, but need not be second-countable. In particular, uncountable discrete groups are, according to this terminology (zero-dimensional) Lie groups and thus in particular pro-Lie groups.

Examples

• Every Lie group is a pro-Lie group.
• Every finite group becomes a (zero-dimensional) Lie group with the discrete topology and thus in particular a pro-Lie group.
• Every profinite 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.
• Every locally compact group has an open subgroup that is a pro-Lie group, in particular every connected locally compact group is a pro-Lie group (theorem of Gleason-Yamabe).
• 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 ${ℝ}^J$ of all real-valued functions of a set $J$ is, with pointwise addition and the topology of pointwise convergence (product topology), an abelian Pro-Lie group, which is not locally compact for infinite $J$.
• The projective special linear group ${\mathrm{PSL}}_2({\mathbb{Q}}_p)$ over the field of $p$-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 satisfies the third condition mentioned above.

References

• Karl H. Hofmann, Sidney Morris (2006): The Structure of Compact Groups, 2nd Revised and Augmented Edition. Walter de Gruyter, Berlin, New York.
• Karl H. Hofmann, Sidney Morris (2007): The Lie-Theory of Connected Pro-Lie Groups. European Mathematical Society (EMS), Zürich, ISBN 978-3-03719-032-6.