Chapter 1: Connected Lie Groups: The Grounding Axioms

Therefore, more formally, I propose the following definition comprising five axioms that, for me at least, is the least redundant, least cluttered, and which also works for general Lie groups whilst admitting techniques found in references like [Rossmann].

Hereafter we shall consider a group $\left(\G,\bullet\right)$. For shorthand we often write the product of two group elements $\gamma_1,\,\gamma_2\in\G$ as $\gamma_1\gamma_2 \stackrel{def}{=} \gamma_1\bullet\gamma_2$.

This group is called a connected Lie group if it fulfills Axioms 1 to 5 below.

Axiom 1 (The “Labeller Axiom” creating a labelled neighbourhood of the identity):

There is some “small” subset $\Nid \subset \G$, the “Fundamental Neighbourhood”, that has a one-to-one “Labeller” map $\lambda : \Nid \to \V\subset \R^N$ (not needfully onto — this will be shown to follow from the axioms 1 to 5 later) defined on it, where the “Namespace” $\V$ is a simply connected, open subset of $\R^N$ (wrt the latter’s standard topology); furthermore $\id\in \Nid$, where $\id$ is the group’s identity element and $\lambda \left(\id\right)=\mathbf{0}$. Here $N$, the dimensionality of $\R^N$, is a finite, positive integer.

We now define the differentiability class of a path.

Definition 1.1 (Differentiability Class of Paths):

A $C^k$ path in $\Nid$ is a map $\sigma : [a,\,b]\to \Nid$ from the real interval $[a,\,b]$ such that $\lambda\circ\sigma\left([a,\,b]\right)$ is a $C^k$ path in $\V$. We say a “$C^k$ path $\sigma : [a,\,b] \to \Nid$ links $\gamma_1 ,\gamma_2\in\Nid$” if and only if $\sigma_\gamma(a)=\gamma_1,\,\sigma_\gamma(b)=\gamma_2$.

With this definition we continue the axioms:

Axiom 2 (The “Connectedness Axiom” defining connectedness of the labelled neighbourhood):

Every element in $\Nid$ can be linked to the identity $\id$ by a $C^1$path.

Witness that the defining of sets $\Nid$, $\V$ and a map $\lambda $ fulfilling the axioms 1 and 2 is nothing special; we can do this for any subset $\Nid$ with the cardinality $\aleph_1$ of the continuum; simply take any set of $C^1$ paths contained within an open ball $\V$ and linked to the origin $\mathbf{0}$ then put them into one-to-one correspondence with any subset of $\Nid$ containing the identity $\id$ and call the inverse correspondence $\lambda$; we need only to define $\lambda(1)=\mathbf{0}$. However, if we do this, the group product and inverse will, in general, shatter the $C^1$ paths by scattering them into disconnected sets. Axioms 1 and 2 become special by holding together with axioms 3 and 4. They are special if the $C^1$ paths they define behave well under the group operations. So the crucial axiom is the following axiom 3; this captures the concept of continuity even though we not as yet defined a topology on $\Nid$.

Notation: $\G$ being a group, then if $\sigma_1 : [a,\,b] \to \G$ and $\sigma_2 : [a,\,b] \to \G$ are paths through $\G$, then $\sigma_3 : [a,\,b] \to \G;\;\sigma_3(\tau) = \sigma_1(\tau) \sigma_2(\tau)$ is also a path through $\G$ and we write $\sigma_1 \sigma_2(\tau)$ as shorthand for $\sigma_1(\tau) \sigma_2(\tau)$. Likewise we write $\sigma^{-1}(\tau)$ for $(\sigma(\tau))^{-1}$.

Now we are ready for the key axiom – the one defining the essence of a Lie group by describing the highly special relationship between the group product and the differentiability class of paths combined by the group product.

Axiom 3 (The “Group Product Continuity Axiom” that group operations preserve differentiable paths):

If $\sigma_1 : [a,\,b]\to \Nid$ and $\sigma_2 : [a,\,b]\to \Nid$ are $C^1$ paths in $\Nid$ and if, furthermore, $\sigma_1(\tau)^{-1}\,\sigma_2(\tau)\in \Nid $ for $\forall \tau \in [c,\,d]\subseteq[a,\,b]$, then $\sigma_1^{-1} \sigma_2 : [c,\,d]\to \Nid ;\,\sigma_1^{-1} \sigma_2(\tau)=\sigma_1(\tau)^{-1}\,\sigma_2(\tau)$ is also a $C^1$ path in $\Nid$.

Axiom 3 packs in a good deal of information. For, in particular, if $\gamma_1 \in \Nid$ and $\gamma_1 \sigma_2([c,\,d])\subset \Nid $ then $\gamma_1 \sigma_2$ is a $C^1$ path (set $\sigma_1(\tau)=\gamma_1^{-1},\forall \tau \in \,[a,\,b]$ in the axiom). Likewise, if $\sigma(\tau)^{-1}\in \Nid $ for $\forall \tau \in [c,\,d]$, then $\sigma ^{-1}: [c,\,d]\to \Nid ;\;\sigma ^{-1}(\tau)=(\sigma(\tau))^{-1}$ is also a $C^1$ path in $\Nid$ (set $\sigma_2(\tau)=1,\forall \tau \in\,[a,\,b]$ in the axiom). Lastly, this axiom also means in particular if $\sigma_1(\tau)\,\sigma_2(\tau)\in \Nid$ for $\forall \tau \in [c,\,d]$, then $\sigma_1 \sigma_2 : [a,\,b]\to \Nid ;\;\sigma_1 \sigma_2(\tau)=\sigma_1(\tau)\,\sigma_2(\tau)$ is also a $C^1$ path in $\Nid$ (write $\sigma_1^{-1}$, which we have just seen to define a $C^1$ path in $\Nid $instead of $\sigma_1$ in the basic axiom). In short, the axiom means that the group product of $C^1$ paths is a $C^1$ path whenever the product stays in $\Nid$, the inverse of a $C^1$ path is also $C^1$whenever it stays in $\Nid$, as are the translations of paths (paths multiplied by a constant group element) staying in $\Nid$.

Now if we were tax lawyers, say, trying to get around Axiom 3, we could construct something fulfilling axiom 3 trivially. Namely, for example, we might define our namespace set to be made up of exactly two $C^1$ paths $\sigma_1,\,\sigma_2$ i.e. $\Nid=\{\sigma_1([a,\,b]),\,\sigma_2([a,\,b])\};\;a,\,b\in\R;\;\id\in\sigma_1([a,\,b]),\,\id=\sigma_1(0)=\sigma_2(0)$ but arrange things such that $\sigma_1(\tau)\sigma_2(\tau)\not\in\Nid\,\forall\tau\neq 0$ so that no part of the product path $\sigma_1\sigma_2$ belongs to $\Nid$ aside from the identity. Thus axiom 3 would be fulfilled vacuously; the question of whether product paths are or are not $C^1$ does not arise. So the next axiom makes sure that we cannot “cheat” in this way.

Axiom 4 (The “Nontrivial Continuity Axiom” that group operations preserve paths nontrivially):

If $\sigma_1 : [-b,\,b]\to \Nid $, $\sigma_2 : [-b,\,b]\to \Nid$ are $C^1$ paths and if $\sigma_1^{-1}(\tau_0)\,\sigma_2(\tau_0)\in\Nid$ for some $-b < \tau_0 < b$ then there is an $\epsilon > 0$ such that $\sigma_1^{-1}(\tau)\,\sigma_2(\tau)\in\Nid\,\forall\, \tau\in[\tau_0-\epsilon,\,\tau_0+\epsilon]$. In words, whenever $\sigma_1^{-1}(\tau)\,\sigma_2(\tau)$ lies in $\Nid$, then $\sigma_1^{-1}(\tau)\,\sigma_2(\tau)$ defines a nonzero length $C^1$ path in $\Nid$ on either side of where $\sigma_1^{-1}(\tau)\,\sigma_2(\tau)$ lies in $\Nid$.

Lastly, we clone the special structure of $\Nid$ and deploy it throughout the whole group by making every patch in the group “locally look like $\Nid$”.

Axiom 5 (The “Homogeneity Axiom” or “Cloning Axiom” that the labelled neighbourhood generates the whole group):

The whole group $\G$ is generated by $\Nid$ i.e. everything in $\G$ is a product of a finite number of members of $\Nid$; $\G=\bigcup\nolimits_{k=1}^\infty\Nid^k$ is a fairly self-explaining shorthand for this idea. Otherwise put, $\G$ is the smallest group containing $\Nid$.

Convention 1.2 (Dimension $N$ in the Labeller Axiom 1):

Hereafter, it will be assumed, unless otherwise said, that the dimension $N$ in Axiom 1 is minimal, in the sense that there is not some smaller dimension $\V^\prime\subset\V$ that is simply connected and open in some lower dimension vector subspace (isomorphic to $\R^M,\,M<N$) of $\R^N$. Sometimes I shall relax that assumption, so I formally add this assumption as a sixth axiom in my next post that I can “switch on and off” for different discussions. The point is that, with the Labeller axiom understood as implying that the dimension $N$ in Axiom 1 is minimal, the above five axioms wholly define a connected Lie group; however, it is sometimes convenient to study entities – matrix subgroups of $GL(N,\R), \,GL(N,\mathbb{C})$ for example) with the assumption relaxed and so that we can “re-use” the $\V$ defined for the full set $GL(N,\R), \,GL(N,\mathbb{C})$ to label our subgroup’s $\Nid$ neighbourhood of the identity. When we do this, I show that there is a natural way to redefine $\V$ and $\lambda$ so that the assumption of minimal $N$ is reinstated.

So there we have it, our five defining axioms for a connected Lie group. Let’s summarise:

  • The labeller and connectedness axioms 1 and 2 set up the basic namespace set and labeller map;
  • The product continuity axiom 3 captures the basic continuity nature of the group product when it applies to paths in the namespace set;
  • The nontrivial continuity axiom 4 makes sure we can’t “cheat” by fulfilling axiom 3 vacuously (trivially);
  • The homogeneity or structure cloning axiom copies and deploys the grounding structure and relationships set up by the other four axioms throughout the whole group;

Let’s sketch what we seem to have so far in the drawing below. For an arbitary group $\G$ we can think of the sketch as the image of $\Nid$ in $\V$. If you’ve been introduced to Lie theory before, the “Spider” (or brittle starfish) we seem to have defined doesn’t look very much like a Lie group, does it? At first glance, it might seem that we have simply a whole gathering of $C^1$ paths, but we shall see in the next section that the drawing below is not really what we have. We shall see that the Continuity Axiom 3 “fills all the spaces between the paths in”. Even though we did not assume in the Labeller Axiom 1 that the $\lambda$ was onto $\V$ (surjective), we shall see that this follows from the axioms. So the image of $\Nid$ is not really a “brittle starfish” at all, but a convex open subset of $\R^N$. So in the end we can take $\V$ to be an open ball which $\lambda$ maps $\Nid$ bijectively onto.

The "Spider" Defined by Axioms 1 through 5

The “Spider” Defined by Axioms 1 through 5

It is worth noticing that these axioms are essentially gotten by standing Satz 1 of [Freudenthal, 1941] on its head: Freudenthal’s theorem notes that any Lie subgroup of a Lie group fulfills axiom 2 (as well as the others by definition). What I shall ultimately show is these axioms are logically equivalent to the modern definition of a Lie group.

Reference:

  1. Wulf Rossmann: “Lie Groups: An Introduction through Linear Groups (Oxford Graduate Texts in Mathematics)”
  2. Hans Freudenthal, “Die Topologie der Lieschen Gruppen als algebraisches Phänomen. I“, Annals of Mathematics, Second Series, 42 #5, 1941, pp1051-1074