Lie groups and Lie algebras – 2

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 \mathbb{R}^n 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 I+\epsilon X 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 T_g G which is the Lie algebra \mathfrak{g}.  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 X\in\mathfrak{g} then e^{tX}\in G 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 \phi:I\subset\mathbb{R}\to G be a path in G.  Also let p be a point on the  path (image of \phi).  We can compose \phi with any chart that p belongs to, say x.  That gives us x\circ\phi :I\to\mathbb{R}^n a path in euclidean space.  We’ll usually write \phi(t)=(\phi^1(t), \phi^2(t),...\phi^n(t)) as the components of \phi 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 \phi and write it as

\frac{d\phi}{dt}-\dot{\phi}(t)=(\dot{\phi}^1(t),...,\dot{\phi}^n(t))

Let a\in [0,1] then at \phi(a)=p\in G we have a tangent vector, called it \vec{v}|_a which is equal to

v|_a=\dot{\phi}(a)=\dot{\phi}^j(a)\frac{\partial}{\partial x^j}

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, T_p G.  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 (g,v_g) where the first coordinate is a point in G and the second coordinate is a vector from T_g G.  TG comes with a natural projection map \pi : TG\to G by (g, v_g)\to g 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, s:G\to TG.  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)

\vec{v}=v^j(x)\frac{\partial}{\partial x^j}

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,

\dot{\phi}(t)=\vec{v}|_{\phi(t)}

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 g\in G we denote the integral curve starting at g generated by vector field \vec{v}, \Psi_g(t).  We call this the flow generated by \vec{v}.  This language is purposefully descriptive.  Since \Psi_g(0)=g we can think of \Psi_g(t) as the image of g under the flow \Psi_g.  The flow takes points and drags them along the manifold such that the tangent vectors to the path of g are those of \vec{v}.

\Psi_{\Psi_g(\delta)}(\gamma)=\Psi_g(\delta + \gamma)

\Psi_g(0)=g

\frac{d}{dt}\Psi_g(t)=\vec{v}|_{\Psi_g(t)}

Thus the flow is a one parameter group action on G.  The group acting is \mathbb{R}.  We call \vec{v} 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 e^{()}? If so, to what extent?

From then on Olver replaces \Psi_g(t)=exp(t\vec{v})g 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 R_g:G\to G be the map defined by right multiplication.  So if h\in G then R_g(h)=hg.  The differential (or pushforward) of this map is  dR_g : T_h G \to T_{hg} G.  A vector field is called right-invariant if for \vec{v}\in T_gG we have dR_g(\vec{v}_h)=\vec{v}_{R_g(h)}=\vec{v}_{hg} 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 \vec{v},\vec{w} and a function f:G\to\mathbb{R} we can define their bracket as [\vec{v}, \vec{w}]=\vec{v}(\vec{w}(f))-\vec{w}(\vec{v}(f)).  In this construction the group manifests itself through the diffeomorphisms D_g 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 v_g=dR_g(v_I).  Though, by this reasoning it would be uniquely defined anywhere, say v_g=dR_{h^{-1}}(v_h).

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, \vec{v}_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 \mathfrak{g} 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 \mathfrak{g} 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.

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About because0fbeauty

Fascinated by the way mathematics and physics interact, captivated by visual and tactile mathematics and hoping to become a better expositor of these things is why I blog...occasionally...when I remember.
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5 Responses to Lie groups and Lie algebras – 2

  1. sheabrowne says:

    I need to read this carefully and compare with my current understanding of tangent vectors and Lie algebras. At first glance this approach looks like the one taken by Wald in his book on General Relativity (with I recommend if you don’t have it). More later, and great post!

    • Please do. I’m still sorting through a lot of the details and trying to reconcile the motivated intuition based ideas I read about in physics with the more formalized approach of mathematics.

      In particular I was thinking about the nature of the tangent space. Consider an n-dimensional manifold, locally it’s homeomorphic (or even diffeomorphic) to R^n but in a far more narrow sense (one I understand less) it is locally linear as well, also in the sense of R^n. There are two ways we’re smuggling in euclidean space, the chart smuggle in differentiable structure and the tangent spaces smuggle in the algebraic structure. What’s interesting to me is that the charts are of finite extend while tangent spaces are pointwise. I’m not sure that makes any sense but it connects to Lie algebras and Lie groups. In any upcoming post I’ll show that the exponential function (of a Lie algebra element) sets up a local diffeomorphism between the Lie algebra and any neighborhood of a point on the Lie group. So the Lie algebra is locally “the same'” as the Lie group. This is analogous to the case of a manifold that is not a group, although with the Lie algebra we ought to be able to recreate the algebraic structure of G, locally, from the algebra g.

  2. sheabrowne says:

    In several places I’m running into the notion of an adjoint representation of a Lie algebra (and in some places even the co-adjoint), and I’m not at all comfortable with this (these) are. For example, if I calculate the adjoint representation of SU(2), I seem to be getting three 3×3 matrices…? Help!

  3. sheabrowne says:

    Ok, it makes sense that I got 3×3 matrices…these are the O(3) matrices that have the same local structure (commutation relations) as SU(2). This seems much deeper than I’m grasping right now.

    • That’s another area I haven’t had a chance to really break into yet: representation theory. I’m afraid I have little to say on this at the moment ;(

      I’m glad you mentioned commutation relations as capturing local structure. I’ve thought about it, but it’s always good to see it written. I really does, doesn’t it since the only data determining a Lie Algebra is base field, dimension and commutator.

      I look forward to reading anything you find out about representations, don’t be shy!

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