### Linear Algebraic Groups (Part 1)

#### Posted by John Baez

I’m teaching an elementary course on linear algebraic groups. The main aim is not to prove a lot of theorems, but rather to give some sense of the main examples and the overall point of the subject. I’ll start with the ideas of Klein geometry, and their origin in old questions going back almost to Euclid.

John Simanyi has been taking wonderful notes in LaTeX, so you can read those!

Here’s the first lecture:

- Lecture 1 (Sept. 22) - The definition of a linear algebraic group. Examples: the general linear group $\mathrm{GL}(n)$, the special linear group $\mathrm{SL}(n)$, the orthogonal group $\mathrm{O}(n)$, the special orthogonal group $\mathrm{SO}(n)$, and the Euclidean group $\mathrm{E}(n)$. The origin of groups in geometry: the parallel postulate and Euclidean versus non-Euclidean geometry. Elliptic and hyperbolic geometry.

It’s sort of spooky how rather old questions about the parallel postulate ultimately led to non-Euclidean geometry and then Klein geometry. When did people start trying to derive the parallel postulate from the other postulates?

And since spherical trigonometry goes back to the Babylonians, why the heck did it take so long for people to notice that spherical — okay, elliptic — geometry obeys all the postulates of Euclidean geometry except the parallel postulate? Was it just the need to switch from the sphere to $\mathbb{R}P^2$?

The idea of alternative geometries should not be all that weird if you spend your nights using spherical trigonometry to study the stars and your days using ordinary trigonometry to study figures drawn on the sand. You might even get the idea that the Earth is a sphere, and that spherical geometry reduces to Euclidean geometry in the limit where the radius of the Earth goes to infinity.

## Re: Linear Algebraic Groups (Part 1)

People have been trying to prove the parallel postulate for a very long time. Proclus in the 5th century already discusses past attempts.

Of course people understood that spherical geometry was different, that it applied to Earth and sky, so in that sense everyone was aware of alternative geometries. But the geometry of the sphere and of the plane were simply different, obviously different. Why would anyone have thought to ask whether spherical geometry satisfied Euclid’s axioms? And if they had, the answer was right at hand: it did not!