October 17, 2016

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.

Posted at October 17, 2016 12:52 AM UTC

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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!

Posted by: Fernando on October 17, 2016 2:16 PM | Permalink | Reply to this

Re: Linear Algebraic Groups (Part 1)

Thanks for pointing out Proclus Diadochus! I didn’t know about ancient Greek (actually Hellenistic) attempts to prove the parallel postulate. Here’s something from his Commentary on Euclid’s Elements:

This [fifth postulate] ought even to be struck out of the Postulates altogether; for it is a theorem involving many difficulties which Ptolemy, in a certain book, set himself to solve, and it requires for the demonstration of it a number of definitions as well as theorems. And the converse of it is actually proved by Euclid himself as a theorem. It may be that some would be deceived and would think it proper to place even the assumption in question among the postulates as affording, in the lessening of the two right angles, ground for such an instantaneous belief that the straight lines converge and meet. To such as these Geminus correctly replied that we have learned from the very pioneers of this science not to have any regard to mere plausible imaginings when it is a question of the reasonings to be included in our geometrical doctrine. For Aristotle says that it is as justifiable to ask scientific proofs of a rhetorician as to accept mere plausibilities from a geometer; and Simmias is made by Plato to say that he recognizes as quacks those who fashion for themselves proofs from probabilities. So in this case the fact that, when the right angles are lessened, the straight lines converge is true and necessary; but the statement that, since they converge more and more as they are produced, they will sometime meet is plausible but not necessary, in the absence of some argument showing that this is true in the case of straight lines. For the fact that some lines exist which approach indefinitely, but yet remain non-secant, although it seems improbable and paradoxical, is nevertheless true and fully ascertained with regard to other species of lines [for example curves like the hyperbola that has asymptotes]. May not then the same thing be possible in the case of straight lines that happens in the case of the lines referred to? Indeed, until the statement in the Postulate is clinched by proof, the facts shown in the case of other lines may direct our imagination the opposite way. And, though the controversial arguments against the meeting of the straight lines should contain much that is surprising, is there not all the more reason why we should expel from our body of doctrine this merely plausible and unreasoned (hypothesis)?

It is then clear from this that we must seek a proof of the present theorem, and that it is alien to the special character of Postulates. But how it should be proved, and by what sort of arguments the objections taken to it should be removed, we must explain at the point where the writer of the Elements is actually about to recall it and use it as obvious. It will be necessary at that stage to show that its obvious character does not appear independently of proof, but is turned by proof into matter of knowledge.

This is fascinating, and leaves me with many more questions.

Posted by: John Baez on October 18, 2016 5:27 AM | Permalink | Reply to this

Re: Linear Algebraic Groups (Part 1)

It’s true that the 5th Postulate is rather wordy, but it’s also the only postulate Euclid gives that says ANY two figures meet in a point.

There’s a notorious story of a young boy writing a competition who decided he needed the fact that “if a line meets one side of a triangle, then it meets one of the other two sides or the opposite vertex,” but “I can’t see how to prove this with Euclid’s axioms”. And that’s also a rather wordy condition. Another axiom assumed but not numbered is: if one circle have center on another circle and radius not more than twice the other, then the circles meet…

Posted by: Jesse C. McKeown on October 21, 2016 10:00 PM | Permalink | Reply to this

Re: Linear Algebraic Groups (Part 1)

Posted by: Jesse C. McKeown on October 22, 2016 2:13 AM | Permalink | Reply to this

Re: Linear Algebraic Groups (Part 1)

Given that Proclus mentions particular substitutes for line which fail to verify the Parallel Postulate (though they also don’t make sense of Postulate 1 — there are many hyperbolas through two points)… and given that he surely knows Spherical geometry… We wonder, in considering his demand for “proof”, whether he means proof that some “geometry” is 5th-Postulate, or proof that the Geon we would Metrein (or its ambient 3-space) is 5th-Postulate? The first case is both reasonable if you have some abstract notion of “a geometry” and (as it turns out) doable; the second… For instance, it is (from the excerpt, at least) unclear whether Proclus expects the Geon or the Cosmos even to be unbounded. (Incidentally, I’ve been told that even Gödel himself believed that, somewhere Beyond, there was an actual ultimate Universe Of Sets, which would have $2^{\aleph_0} = \aleph_2$… well, I believe there is an actual Natural Numbers… So.)

Are these like some of your wonderings?

Partly in reply to Fernando, It is also unclear (at least from the excerpt) whether Proclus or his fellows had a conception of Spherical Geometry as what we now call an “intrisic geometry”, or perhaps only as a particular figure in 3-space readily available for study whether navigating or stargazing.

Posted by: Jesse C. McKeown on October 24, 2016 3:22 AM | Permalink | Reply to this

Re: Linear Algebraic Groups (Part 1)

Just pointing out a small typo in the pdf: in the last line on p.2 I think the last negation shouldn’t be there (in the clause stating that the two lines are parallel).

Posted by: Isabel on October 27, 2016 2:01 PM | Permalink | Reply to this

Re: Linear Algebraic Groups (Part 1)

Thanks! I’ll check it out and fix it if necessary.

Posted by: John Baez on October 27, 2016 8:35 PM | Permalink | Reply to this

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