Why Yet Another Exposition on Lie Groups and Lie Theory?

The modern definition of a Lie group is a topological group that is also an analytic manifold such that the group operation is analytic in the coordinates used to label the manifold. [Spivak] teaches Lie groups like this in the first volume, for example. There is nothing wrong with this definition, of course, but I don’t believe this one is the best one for teaching or learning the fundamentals of Lie groups.

A Lie group is a very special kind of manifold: a manifold is far too broad and general a thing and you do not need anything like the whole of differential geometry to understand a Lie group well. Although it is good to abstract and generalize, you might say that this approach of learning differential geometry first and then going onto tackling Lie groups is looking at the trees of the forest from too far away for a good first look at the trees.

One way wherein a Lie group is special is that its fundamental group is Abelian (and we’re talking here about a connected group, so that we can meaningfully talk about “the” fundamental group), as is indeed the fundamental group of any connected topological group. In contrast, for every finitely presented free group $G$, there is a connected analytic manifold $M_G$ with this group as its fundamental group, i.e. $\pi_1(M_G) = G$. A general analytic manifold is therefore a much more complicated thing than a Lie group.

A second specialisation is that, through the operation of left and right translation of the Lie algebra, one can easily define vector fields which are nowhere vanishing and everywhere continuous on a Lie group: these are the so called left- and right-invariant vector fields on the Lie group. Therefore, there is no Lie group whose manifold is homeomorphic to an even dimensional n-sphere, because the hairy ball theorem (see, for example, [Jarvis & Tanton, 2004]) forbids such a vector field on the latter. Thus we have the one-sphere $U(1)$ (group of unit magnitude complex numbers) and the three-sphere $SU(2)$ (group of unit quaternions) as Lie groups, but the only two-dimensional compact Lie group is a torus. A Lie group’s hair can always be combed everywhere flat.

Thirdly, with these nowhere vanishing, everywhere continuous left/right-invariant vector fields, which span the whole tangent space at every point in a Lie group, we can define a notion of parallel transport that is independent of path – this situation is, of course, atypical in a general differential manifold, where the holonomy group is typically nontrivial. That is, although a general Lie group is most decidedly not linear, it can be given a flat connexion. The torsion, of course, is nonzero to encode the Lie groups general non-Euclideanhood.

The left/right-invariant vector fields as well as many other special properties peculiar to Lie groups arise from the more general notion of homgeneity (which holds in the more general notion of a topological group, of which a Lie group is one important example). If a neighbourhood $\mathcal{N}$ of the identity $\mathrm{id}$ of the group $\mathfrak{G}$, and if $\gamma\in\mathfrak{G}$ is any group member, then the set $\gamma \mathcal{N}$ is a neighbourhood of $\gamma$. Thus a Lie group looks locally the same from whatever standpoint $\gamma$ one may look at it from. It is this homogeneity that allows one to infer some global topological behaviours from the Lie algebra, i.e. from local properties. For a Lie group with a finite centre is compact if and only if its Killing form is negative definite, as shown in [Helgason].

Clearly we are dealing with quite a specialised and restricted idea of a manifold here.

My approach is to make the notion of a one-dimensional, once differentiable ($C^1$) path or “space curve” the fundamental building block for building a Lie group with. The Lie group is then a group which has at least one neighbourhood $\mathcal{N}_{\rm id}$ of the group identity ${\rm id}$, wherein each point in the neighbourhood is linked to the identity by a $C^1$ path. To make the concept of $C^1$ meaningful, there must be a “labelling map” $\lambda:\mathcal{N}_{\rm id}\to \mathbb{R}^N$ that labels each member of $\mathcal{N}_{\rm id}$ by unique coordinates in $\mathbb{R}^N$: a $C^1$ path is then one which when mapped into $\mathbb{R}^N$ is $C^1$ with respect to the wonted topology for $\mathbb{R}^N$. Note that I don’t assume that the image $\lambda(\mathcal{N}_{\rm id})\subset \mathbb{R}^N$ is even open in $\mathbb{R}^N$: this actually follows from my axioms to be presented. Once we’ve defined one neighbourhood like $\mathcal{N}_{\rm id}$, we define the Lie group as the smallest group $\mathfrak{G}$ containing $\mathcal{N}_{\rm id}$.

The Lie group thus builds itself from very simple axioms. And it does so in a way which needfully means it is analytic ($C^\omega$), not simply smooth. The simple $C^1$ assumption implies $C^\omega$. Once the Lie group has built itself for us, we can use it as an object of study to get intuition for the differentiable manifold concept. The theory of Lie groups can thus be an intellectual steppingstone towards learning differential geometry, not the other way round.

Indeed Montgomery, Gleason and Zippin’s solution to Hilbert’s fifth problem goes even further and shows that not even differentiability needs to be assumed, for it emerges naturally from only the assumptions about a Lie group’s continuity. Yet even this amazing theorem is not all in some cases. The Lie group idea emerges from even more primitive assumptions in the case of compact semisimple Lie groups: for there is no other abstract group structure possible for such a Lie group so even the topology emerges from the algebraic structure alone and every group automorphism as an abstract group preserves the Lie group structure as well [VanDerWaerden]. Lie groups are thus seen to be a very special kind of manifold indeed and the modern idea of a manifold contains too much machinery to see them clearly with. Intuitively, this highly special nature comes from homogeneity – the fact that the group action clones the around-identity-neighbourhood structure throughout the whole manifold and there just aren’t that many axiom systems and behaviours that can withstand such wholesale “cloning” and still be consistent.

References:

  1. Spivak, M, “A Comprehensive Introduction to Differential Geometry”, 3rd edition, Publish or Perish, 1999
  2. Tyler Jarvis and James Tanton, “The Hairy Ball Theorem via Sperner’s Lemma”, Amer. Math. Monthly, 111, No. 7 2004, pp. 599-603
  3. Helgason, S., “Differential geometry, Lie groups and Symmetric Spaces”, Chapter 2, §6, proposition 6.6.
  4. van der Waerden, B. L., “Stetigkeitssätze für halbeinfache Liesche Gruppen” (Continuity Theorem for Semisimple Lie Groups), Mathematische Zeitschrift 36 pp780 – 786