Informal Definitions for the Lie Group Axioms

To define a continuous group, we need a one-to-one map, which I shall call the “Labeller”, that gives unique co-ordinates (in $\mathbb{R}^N$) to every element of the group that is “near enough” (yet to be defined) to the group’s identity element. Specifically, there is a particular subset, which I shall call the Namespace1, comprising all the elements that are named in this way. This “naming” preserves information by dent of its being one-to-one throughout the Namespace, so that we can study the group product and inversion operations simply by looking at these operations’ effect on the co-ordinates (“names”), even though the group elements themselves might be any kind of weird creatures. We call the group a Lie group if we find that these equivalent “stand-in” product and inversion operations on $\mathbb{R}^N$ are continuous and differentiable and, furthermore, that the first derivatives are also continuous. For Lie theory then, the two essential roles fulfilled by the Labeller and Namespace are that:

  1. They bestow a topology, i.e. a concept of neighbourhood and nearness, on some “small set” near the identity through the standard topology in $\mathbb{R}^N$ defined on the co-ordinates;
  2. They attach a “frame” to the identity, so that tangents and derivatives can be measured. In a general Lie group as opposed to a linear, matrix Lie group, a difference operation between group elements may not even be defined, so we must measure rates and directions of change by looking at the differences between the co-ordinates in $\mathbb{R}^N$ corresponding to elements.

This minimalist approach is more like the “local” approaches taken in the early days of Lie theory or indeed somewhat like that taken by Montgomery, Gleason and Zippin in their (difficult) answer to Hilbert’s fifth problem, i.e. roughly, that the differentiability class of the group operations does not matter; we don’t even have to assume differentiability and an assumption of only continuity ($C^0$ group operations) will do. Our needs for now are much less lofty and I shall assume $C^1$ group operations. We still have the satisfying fun of seeing a lowly assumption of $C^1$ implying the seemingly hugely stronger $C^\omega$ (much as a $C^1$ function of one complex variable is needfully holomorphic, $C^\omega$) and, moreover, it turns out that we can take an even simpler approach than the above by thinking of only one-dimensional paths in the group defined by $C^1$ paths in the co-ordinate space $\mathbb{R}^N$. We can then define a Lie group as one whose group operations preserve the $C^1$ differentiability class of these paths2: if $\sigma _1 ,\;\sigma _2$ are $C^1$ paths, then so are $\sigma _1 \bullet \sigma _2$ and $\sigma _1^{-1},\;\sigma _2 ^{-1}$. This approach lends itself to techniques most like those in references such as [Rossmann] that make the fundamentals of Lie theory so clear and readily grasped; unlike [Rossmann] it is not restricted to linear (matrix) groups. Whilst proving that this approach leads to the same concept of Lie group as the standard definition (i.e. as a group that is an analytic manifold and whose group operations are $C^\omega$) we shall see that the analytic manifold “builds itself” from our definition.

Notes:

  1. I have stolen this word from computer science, where it is defined as “an abstract container or environment created to hold a logical grouping of unique identifiers or symbols (i.e. names)”. The Namespace contains the group’s identity ${\rm id}$. Although the Labeller gives “names” to group elements that are unique within the subset; we shall see that this subset and co-ordinates are used again over and over for different regions in the group.
  2. We can equivalently, and more pithily, state that if $\sigma_1,\;\sigma _2 $ are $C^1$ paths, then so is $\sigma _1 ^{-1}\bullet \sigma_2$; this is a standard trick to consolidate group axiom systems – we don’t then have to talk about products and inverses separately preserving continuity or differentiability classes.

Reference:

  1. Wulf Rossmann: “Lie Groups: An Introduction through Linear Groups (Oxford Graduate Texts in Mathematics)”