From: kramsay@aol.commangled (Keith Ramsay) Subject: Re: Lie Groups Date: 02 Jul 2000 16:39:20 GMT Newsgroups: sci.math Summary: [missing] In article <8jdh90$oea$1@news.panix.com>, erg@panix.com (Edward Green) writes: |What are the defining features of a Lie group, as opposed to any other |group? In a Lie group the elements of the group are points of a smooth manifold, and the group multiplication and taking of an inverse are smooth maps. An ordinary function f(x1,...,xn) of several variables is called "smooth" if we can take partial derivatives any number of times and they are all defined whereever f is. So for example, f(x)={0 if x<0; x^2 if x>=0} is not a smooth function because although we can take its derivative once and get {0, if x<0; 2x if x>=0}, we can't take its derivative a second time at x=0. But the most familiar functions are all smooth where they're defined: the four basic operations, trig functions, exponentials and logs, functions you can make by combining them, and so on. A manifold is a (topological) space which is covered with coordinate charts; each chart associates each point in a region of the manifold with a point in a region of n-dimensional coordinate space. The association has to be 1-1; each point of the region of the manifold covered by the chart gets assigned a unique set of coordinates (x1,...,xn) (where the manifold is n-dimensional), and each point in a region of coordinate space is associated to a point of the manifold. Where two coordinate charts overlap, we can consider the function converting one set of coordinates with the other one. For a vanilla manifold, these overlap maps are required to be continuous functions. In order for a manifold to be "smooth", the overlap mappings have to be smooth functions. In order for the group multiplication to be "smooth", it has also to be defined by smooth functions. So basically a Lie group is a group where the multiplication* is defined by smooth functions. It's just that it only makes sense to talk about "smooth" functions if we're on a manifold. This property allows us to use techniques from calculus and geometry, while the group structure by itself is thoroughly algebraic. [*] And taking of inverse. There's a relatively useless way to make any group into a Lie group: use the topology in which each point is isolated from all the others, and consider it a 0-dimensional manifold, in which each point is in its own coordinate chart where it's assigned coordinates (). What makes a Lie group interesting as a Lie group is actually the structure of its connected pieces. So it's fairly common to consider only connected Lie groups. The rational numbers Q considered as a group under addition isn't a connected Lie group because it's not a manifold. We can put a coordinate chart of a kind on it, but not every coordinate associates with a point of the group. It's "missing" too many points. One of the passages in mathematics which I consider prettiest is the line starting from the definition of Lie group and leading to some standard structure theorems for Lie groups. At first it seems we only have some fairly amorphous object without a lot of properties, but we gradually discover more structure on it. It turns out that we don't actually need all kinds of smooth functions to define them; as I recall it turns out we only need polynomial functions (although I may be forgetting some technical condition required to get to this). Having smooth coordinate charts allows us to use one of the most powerful techniques, which is to consider the tangent plane to the Lie group at the identity element, which is called the associate "Lie algebra". If you don't feel the need to be more than twice as precise as physicists usually are, it's quite okay to think of Lie groups as always being groups of invertible matricies defined by some polynomial conditions, with the multiplication being simply matrix multiplication. For instance the invertible 2 by 2 matricies (a b) (c d) form a group called GL(2,R), R for real numbers. Within GL(2,R) we have the subgroup defined by a=d=1, c=0, which is isomorphic to the group of real numbers under addition, since (1 b)(1 b')= (1 b+b') (0 1)(0 1 ) (0 1 ). That's the connected 1-dimensional Lie group. If we require only a=1 and c=0, we get a not very complicated 2-dimensional Lie group often called the "ax+b" group because it's equivalent to the group of transformations of the form x --> ax+b, where in order to be invertible we need a<>0. That breaks into two connected components, where a>0 and where a<0. Lie group theory is one of those happy areas where mathematicians had already developed a lot of the theory when physicists realized they needed it. Keith Ramsay ============================================================================== From: baez@galaxy.ucr.edu (John Baez) Subject: Re: Lie groups for grunts Date: 18 Oct 2000 08:43:00 GMT Newsgroups: sci.physics.research In article <8scnp3$cpb$1@bob.news.rcn.net>, Michael Weiss wrote: >Here are some things I think are true, >either because they *ought* to be true, or because maybe once upon a time I >read something like it somewhere. Okay, let's see if I can whether they ARE true. >1) If we have a group which is also a differential manifold, and the group >operations are smooth (C^infinity), then in fact they are analytic. And >probably C^infinity is even too strong; my guess is that C^3 will do. This is a famous issue - it's one of the problems Hilbert posed in 1900. Reduce to a bare minimum the assumptions needed for a topological group to be a Lie group! In Lie's original definition, all the group operations needed to be *analytic* functions. But having them be *smooth* (i.e. infinitely differentiable) is enough to imply they are analytic. And in fact, we can get a way with a lot less than that. In 1888, Lie claimed that having them be *differentiable* is enough to imply they are analytic. But Lie didn't provide a proof - and with a name like that, who would believe him? Luckily, Schur provided a proof in a series of papers published in 1890-1893 - and with a name like that, we can be sure it's true. So, in 1900, Hilbert went wild and asked if we can get away with having the group operations be *continuous*. In modern lingo: given a topological group that's a topological manifold, does it always have a unique real-analytic structure so that the group operations are analytic??? The answer is YES! After many partial results, this was proved by Gleason, Montgomery and Zippin in 1952. >2) There are infinite-dimensional Lie groups, which of course are not >isomorphic to matrix Lie groups. Yes, and they are very important in physics, but now of course you're stretching the term "Lie group" - an "infinite-dimensional Lie group" is like a "nonassociative algebra": the first modifier is taking away properties we normally expect. To be precise we need to talk about "Hilbert Lie groups", or "Banach Lie groups", or other specific sorts of infinite-dimensional analogues of Lie groups, corresponding to various specific sorts of infinite-dimensional manifold. 3) Every finite dimensional Lie group is isomorphic to a matrix group. No! It ought to be true, but it's not: the math gods were in a bad mood that day. Take the universal cover of SL(2,R). This has no faithful finite-dimensional representations, so it's not isomorphic to a matrix group. At times I have known the proof of this, but now I don't, so don't ask me why it's true, unless you wish to make me think. If we restrict attention to *compact* Lie groups, we're okay: they are all isomorphic to matrix groups.