What can be said about Lie groups in a first abstract algebra course?

I suggest having a look at the following book by John Stillwell: http://rads.stackoverflow.com/amzn/click/144192681X. 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, http://rads.stackoverflow.com/amzn/click/1133599702, 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.

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.

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.