A book I’ve referred to in the past and have returned to is Olver’s book on Lie groups. It’s very conversational and focuses on many examples and calculations. Reading it again recently has helped connect a couple of ideas that I’ll try to explain here.
Olver’s construction of the Lie algebra is different than Gilmore’s and we might categorize them as ‘mathematical’ and ‘physical.’ Let me summarize the last post Lie groups and Lie algebras – 1 which was largely motivated by Gilmore’s book.
We start with a Lie group G and focus on a neighborhood of the identity. A Lie group is locally diffeomorphic to for some n. So consider, heuristically, a point, g, close to the identity I. That point can also be viewed as an operator (for some representation of the group) and so we can expand it as a power series and drop all but first order terms in the perturbation obtaining something like where X is the derivative(s) of the operator with respect to the local coordinates. These will end up being elements of the tangent space which is the Lie algebra . The group operation and the group commutator will induce the structure of an algebra onto this tangent space. So in this sense, it’s because we’re tangent to a group that the tangent space becomes an algebra.
Because any group element close to I can be expressed, perturbatively, we can, through the group operation, generate other group elements. It turns out that this corresponds to exponentiation of elements of the Lie algebra. So if then where t is a real number. (note: lots more about the exp function in upcoming posts)
The other approach, one that appears in Olver’s book, is to construct the Lie algebra from right invariant vector fields and then to show that this construction is isomorphic to the tangent space of the identity element of G. Along the way we’ll find some alternative (and more rigorous) explanations behind some of the common expressions I’ve seen in physics texts.
I’m due to write a sequel to Tangent Spaces – 1 where I work out another method of constructing the tangent spaces. I’ll briefly mention it here, see Olver for more detail. The idea is to consider paths in G. Let be a path in G. Also let p be a point on the path (image of ). We can compose with any chart that p belongs to, say x. That gives us a path in euclidean space. We’ll usually write as the components of and think of these as components in G rather than n-space. Very soon I’m going to have to blog a number of examples to cement this nebulous understanding I have. On the one hand it’s easy to differentiate this composition since it’s just a vector valued function from [0,1]. Computing tangent vectors to this curve is nothing more than third semester calculus material. On the other hand we’re also doing this on a group but I suspect this sounds more complicated than it need be. In any event, we can differentiate and write it as
Let then at we have a tangent vector, called it which is equal to
Where Einstein summation is in effect and the partials are with respect to the current chart. This is one tangent vector at the point p and we can obtain all of the others in the same way. Consider all paths with p in their image and consider the derivatives of those paths at p. All those vectors together form the tangent space at p, . There’s a bit more to this construction that involves some equivalence classes that I’ll get to in the next post on Tangent Spaces.
So to Olver, a tangent space is a collection of tangent vectors, more intuitive than derivations certainly. Still, it’s important to remember that unlike tangent vectors to a curve in euclidean space these do not ‘stick out’ off the surface or curve. There’s nothing outside the manifold G.
If you take all the tangent spaces for all the points on G you can obtain the tangent bundle, TG which has a topology and differentiable structure induced on it from G. TG is 2n-dimensional and has points where the first coordinate is a point in G and the second coordinate is a vector from . TG comes with a natural projection map by so it maps entire tangent spaces (fibers) to a point. The point of introducing the tangent bundle is to talk about vector fields. You might remember from the post on Pullback Bundles and from Spheres-2 some details about fiber bundles. Along with the projection there’s a map called a section which is a right-inverse of the projection map, . If s is smooth then it represents a vector field on G. Basically s assigns to each point of G, a vector from it’s tangent space but we want this done in a differentiable manner not willy nilly.
So a vector field on G looks like, (in a local chart x)
This should look familiar as we had something like this a few paragraphs before, that was the derivative of a path. We used paths to define tangent vectors and now we turn the idea on it’s head and define an integral curve of a vector field. Simply, it’s a curve such that it’s tangent vectors at each point agree with those of the field,
This is another justification for using differential operators as tangent vectors. Derivatives are inherently local statements. If we know the derivative at many points (which means knowing how the function is changing, which amounts to a vector field) when we can reconstruct the unknown function piece by piece (slope fields are a good visualization of this process). It turns out that these integral curves have much more structure than simply being solutions to differential equations, they are one parameter groups of diffeomorphisms of G.
Given we denote the integral curve starting at g generated by vector field , . We call this the flow generated by . This language is purposefully descriptive. Since we can think of as the image of g under the flow . The flow takes points and drags them along the manifold such that the tangent vectors to the path of g are those of .
Thus the flow is a one parameter group action on G. The group acting is . We call the infinitesimal generator of the action. Again, differential equations are statements of a local nature, it’s not uncommon to think of a derivative as the infinitesimal change in a function with respect to the variable. So, speaking loosely in the language of calculus, vector fields are differential equations and flows are the solutions.
Olver then says that the above properties are reminiscent of the exponential function, so we refer to the above process as exponentiating the vector field. There are more analogies as well as we’ll see in future posts if Shea doesn’t beat me to it. Though I think the idea is not to take it too seriously which is why Olver only uses the symbol “exp” instead of “e.”
Question: does it make any sense to think of the exp() formal function as ? If so, to what extent?
From then on Olver replaces so we have the flow, now denoted by the exponentiated vector field, acting on the element g of G. I’ll talk more about vector fields, flows and 1-parameter groups later. For now let’s get on with constructing the Lie algebra for G.
We start by defining diffeomorphisms that correspond to each element g. Let be the map defined by right multiplication. So if then . The differential (or pushforward) of this map is . A vector field is called right-invariant if for we have for all h and g in G. These vector fields will form a vector space and this vector space is the Lie algebra of G.
This construction is a bit different as it doesn’t require any perturbative expansions. The bracket on the space is the Lie bracket you can put on vector fields in general. Given two vector fields and a function we can define their bracket as . In this construction the group manifests itself through the diffeomorphisms which correspond to points in G. The idea of right-invariant vector fields aren’t special to Lie groups. I suspect you can just consider the group of diffeomorphisms on a manifold M and then look at which vector fields are invariant (in the sense above). It would seem you could make something like a Lie algebra, could you?
Question: Given a manifold M and a group of diffeomorphisms acting on M, what do you get if you repeat the above construction?
How does this connect with the identity element of G? Every vector field is uniquely determined by it’s value at the identity since . Though, by this reasoning it would be uniquely defined anywhere, say .
Question: is this all that particular? Considering vector fields as differential equations which are defined locally anyway and yield global results. Certainly the above point is made using the group structure of G, but does it have to?
The converse holds as well, given a vector at the identity I, determines a right invariant vector field on G as well. In this way there’s a one to one correspondence between the Lie algebra and the tangent space at I. However, all the tangent spaces of G are isomorphic, so this also isn’t very amazing.
I want to finish up with a result. I have a talk coming up in a week and a half so be prepared for some posts on Penrose tilings. Also I need to take some time and start doing calculations for some of this stuff before I try to digest any more. However, before we go here’s a tantalizing tidbit.
1-dimensional subspaces of correspond (1-to-1) with 1-parameter subgroups of G. There’s a lot in that to unwind and examples will help. This is a really important result. Basically the basis elements of can be exponentiated to flows and these flows are diffeomorphisms on G by G. This process of using infinitesimal generators (elements of the Lie algebra) to generate diffeomorphisms happens all over a physics. One of the posts I want to work on soon is analyzing the Lie group of symmetries of Euclidean spacetime (nonrelativistic), E(3) which is the slow cousin of the Poincare group. It’s from that analysis that much of nonrelativistic quantum mechanics will fall out.