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Lie groups are among the most important examples of groups in mathematics and physics, but they are rarely discussed in introductory undergraduate abstract algebra courses, which tend to focus on finite groups.

Partially, this is because you can't even define a Lie group without knowing what a differentiable manifold is, which requires some amount of differential geometry or topology. In addition, most of the main results about Lie groups involve the notion of a Lie algebra, which would require a significant detour to develop.

So my question is, can anything coherent be said about Lie groups in a first abstract algebra course? For example, if I were willing to spend a week or so discussing topics related to Lie groups, what could I cover that would be meaningful? Does anyone have any experience including material along these lines?

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    $\begingroup$ I don't have experience teaching the abstract algebra course, however, I do manage to essentially present the manifold structure of matrices in my advanced calculus course. Given that backdrop, matrix Lie groups aren't that bad. Moreover, books exist. I have several, but, I'm still refining the optimal path. Certainly, the matrix exponential is key. $\endgroup$ – James S. Cook Apr 7 '14 at 5:08
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    $\begingroup$ Hmm. I have made my student calculate the matrix exponential of $$t\pmatrix{0&1\cr-1&0\cr}$$ to make them see something familiar pop out. But this was in a freshman calculus course in the Spring while studying power series (so they've had a semester of calculus and linear algebra prior to that). You could extend this to an explanation of the infinitesimal generator for the group of rotations in the plane (or in 3D about a given axis). I haven't thought about ever going beyond that. Admittedly in the HW assignment of my course this week we introduce the quaternions, which opens up another road. $\endgroup$ – Jyrki Lahtonen Apr 7 '14 at 7:02
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    $\begingroup$ As far as the definition of a differentiable manifold goes, what about settling for topological groups, leaving the existence of some of the real-analogous phenomena an unsettlingly convenient but predictable consequence of taking natural examples of manifolds that are also groups? I have in mind the circle group and group of unit quaternions. If you can define the Lie bracket of smooth vector fields, the abstract algebra subsuming it can be shown, and on to the structure. Learning to speak the language of Lie groups is quite separate from how to identify them, the part you can teach yourself. $\endgroup$ – Loki Clock Apr 7 '14 at 13:19
  • $\begingroup$ Alternatively, you could talk about "matrix groups and their quotients", which probably covers all the examples you want. Then the differentiable structure comes for free from the matrix coordinates. $\endgroup$ – user173 Apr 7 '14 at 14:29
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    $\begingroup$ I wonder to what extent thinking of it as a physicist does would help? I say "I wonder" because I have never understood the "… for physicist"-type books well enough to say for sure, but it seems to me that a physicist's approach to Lie groups is very much through the sort of intuition that you'd like to help students foster, rather than through the sorts of formal definitions that require heavy prerequisites. $\endgroup$ – LSpice Aug 29 '14 at 23:53
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It is possible to usefully mention "Lie groups (and Lie algebras)" in an introductory course, if one does not give formal definitions, but, rather, examples. It is not necessary (or advisable) to "define" smooth manifolds, which seems to have considerable baggage-of-abstraction of its own. Just give important examples, noting that they do seem to have a lot in common: circle group, $SU(2)$, $SO(3,\mathbb R)$, $SL(2,\mathbb R)$, letting them act on spheres, and maybe by the "linear fractional transformation" action on the upper half-plane?

The Lie algebra and exponentiation are easy to do in the case of linear Lie groups, and illustrate many interesting points... and the Lie bracket presents itself as an artifact.

My experience indicates that such a discussion will stimulate some students, and, as usual, upset those students who are already barely keeping up. All the worse that the feasible style cannot be as formal (or else the start-up cost becomes prohibitive), and, ironically, getting away from the usual exaggeratedly formal style can be psychologically disturbing to some students, if only because it's a change from the usual.

But introducing such ideas is a positive thing, and serves very well any students who will be continuing seriously in mathematics or any science that uses serious mathematics. For that matter, those who might have been bored and disenchanted because the all-too-common sterilized-formal abstract algebra was too easy may be provoked a bit, in a good way, by the interaction of more than one idea at a time.

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I suggest having a look at the following book by John Stillwell: Naive Lie Theory. A fast-paced week or (if you're lucky enough... or believe it's worth it as I would) two weeks could be created by pulling the big ideas from chapters 1, 2, and 4.

I would argue that this shouldn't be the first time the students are seeing groups like SO, SU, etc. Those should be introduced as soon as possible as examples of different groups in the beginning of the course (see, for example, Gallian, chapter 2). Then as you near the end of your tour of group theory, you could introduce the ideas from Stillwell which would bring in concepts from Calculus (tangent space!) that might really interest those future mathematicians of yours. How often do you get to see analysis and algebra interacting at these stages?! Handwaving is your friend here, in my opinion. This is a sampling of the future awesomeness.

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  • $\begingroup$ I was looking today through one book and remembered this question. If you look at Algebra (Michael Artin), chapter 8 -- he spends the chapter dealing with linear groups, even getting to the Lie Algebra (his entire text relies heavily on Linear Algebra as motivation, which I like). $\endgroup$ – Zach Haney May 19 '14 at 1:43
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I think in a first abstract algebra course the goal is simply to make students aware that such things exist, give a couple of examples, and let them know that there is much, much more that can be learned in future classes. With that in mind:

Start with $SO(2)$. Note that all elements of $SO(2)$ can be represented in terms of a continuously-varying parameter $\theta$, but not uniquely, and use that fact to interpret $SO(2)$ as the points on a circle. Focus on two key ideas: (i) The group contains infinitely many elements, but is unlike the "discrete" examples (like $\mathbb{Z}$) that they already know about because of its intrinsic "continuity" (I would not formally define or prove anything about the continuity but simply rely on the intuition that circles are smooth curves, and leave it at that.)

Second example: $U(1)$. Give basic definition, interpret it geometrically as a circle, and then presto, $U(1) \cong SO(2)$.

Third and fourth examples: $SO(3)$ and $SU(2)$. The only goal should be to show that both groups can be parametrized by four continuous variables, and that those parameters satisfy the equation of a 3-sphere. Any more than that is overdoing it.

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Good opportunity to introduce some classic arrays in combinatorics and the special linear algebra and group (sl2, SL2) to aspiring mathematical physicists maybe as an extra-point homework discovery process:

1) How would you represent in terms of matrices the action of the derivative $D=\frac{d}{dz}$ on functions represented as power series $f(z) = a_0 + a_1z + a_2 z^2 + ...$ about the origin, i.e., in terms of a matrix action on the coefficient vector representing the power series in the power basis?

2) Same for the higher derivatives.

3) Same for $\exp(tD)$.

4) How would you represent the action of $\exp(tD)$ on $f(z)$ in terms of a transformation of its argument represented as $ M · [z] = \frac{az+ b}{cz + d}$ with $M$ a two-dimensional matrix?

5) Demonstrate the group properties of these reps.

6) Repeat the same process for $zD$ and $z^2D$ noting in general that $g(z)D = \frac{d}{dh(z)} = \frac{d}{d\omega}$ for $\omega = h(z)$ and $z = h^{-1}(\omega)$.

7) Do the reps of $\exp(t_nz^{n+1}D)$ for $n=-1,0,1$ form a group together?

For futher study, see Wikipedia on linear fractional (Moebius) transformations and the Witt Lie algebra and Needham's "Visual Complex Analysis". Also look at the reps of $(zD)^n$ and $z^{-n}(z^2D)^n$ expressed as polynomials in $x^mD^m$ for $m \le n$. Use the OEIS to identify these polynomials.

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