Sporadic discussion

Linear factors for the action of an algebraic group on a split unipotent group

Submitted for publication.

Abstract: Let \(k\) be an arbitrary field, let \(G\) be a linear algebraic group over \(k\), and let \(V\) be a vector group over \(k\) on which \(G\) acts by automorphisms of algebraic groups. The action of \(G\) on \(V\) is said to be linear if there is a \(G\)-equivariant isomorphism of algebraic groups \(V \simeq \operatorname{Lie}(V)\). We give examples of vector groups \(V\) having non-linear action of \(G\). On the other hand, if the \(G\)-module \(A(V)\) of additive regular functions on \(V\) is completely reducible, we show that the action of \(G\) on \(V\) is linear. Using this result, we prove that if the unipotent radical of \(G\) is defined and split over \(k\), then any split unipotent algebraic group \(U\) over \(k\) on which \(G\) acts by group automorphisms has a filtration by \(G\)-stable closed subgroups \(U = U^0 \supset U^1 \supset \cdots\) for which each successive quotient \(U^i/U^{i+1}\) is a vector group having a linear action of \(G\).

This quiz did fairly well – I am from the south, though not from "the midlands".

• 2008 Fundamental lemma seminar of Hales
• Franz Lemmermeyer has a large compilation of links for lecture notes.
• David Ben-Zvi's page Intro to geometric Langlands
• Paul Smith's page Noncommutative geometry and algebra

Here are the slides from my talk in the special session on algebraic groups (etc) at the Joint Math Meeting in New Orleans, Jan 2011. The talk discussed the following results: if G is a linear algebraic group with "nice enough" unipotent radical, and if G possesses a Levi factor after scalar extension to a separable closure of the ground field, then G already has a Levi factor defined over the ground field.

Here is a graph paper generator available on the web – a handy (?) way to play about with your facets for affine A2 or B2 Weyl groups.

I'm glad that Polo and Gille are (hopefully still) producing a new edition of SGA3.

Here I'm including some links for the PDFs of slides for several recent talks.

• Ascona slides, August 2009 These slides are from my talk at the conference on Algebraic Groups and Invariant Theory in Ascona Switzerland, August 2009.
• Raleigh special session talk from the AMS sectional meeting in Raleigh of April 2009.
• slides for my seminar talk at UMich of Dec 2008.
• slides from my talk in Kalamazoo from the AMS sectional meeting in Kalamazoo MI of Oct 2008.

So I'm several pages into Against the Day. And I've discovered the Pychon Wiki and the Chumps of Choice weblog – these appear to be good resources… The moniker "Chumps of Choice" being some deformation of "Chums of Chance", I suppose.

My reading begins shortly after a second reading of Mason and Dixon (I first read M&D in 1999, I believe). And I now wish I had taken notes, or at least scribbled thoughts during that second reading. So I'm hoping to do better with AtD.

Somehow I've missed most of the reviews. I read one in the Boston Globe perhaps in November – it was alright but seemed to me to overplay the identification of Pynchon with some sort of "sixties-ness"; that identification seems to trivialize matters, and it grates on me. I enjoyed the review by Michael Moorcock that I just found – linked by a poster to the Chumps' blog. That review appeared in the Daily Telegraph, and in part MM writes:

Against the Day is a fine example of successful marriage between the popular and the intellectual, between fiction and science. … Aldiss, Burroughs, Ballard and Vonnegut predicted, long ago in the 60s, that the arts and sciences would be reunited in speculative fiction, that the novel would not die if it could rediscover vulgarity.

Gloriously, demandingly, daringly Pynchon has rediscovered vulgarity…

I just finished reading the book: Rising Tide: The Great Mississippi Flood of 1927 and How It Changed America, by John M. Barry. The title more-or-less tells you what the book was about, of course. It was an interesting read; it discusses a number of – sometimes suprising – topics.

The greater-Boston area turns out to be a reasonable place to Diversions. On Sunday, I did a nice ride…

On the other hand, the street sign situation here is pretty frustrating: there seems to be a "New England custom" of not (always) regularly identifying main streets. Exaggerating a bit, I guess the thinking is: if you don't know the name of the street, you shouldn't be on it.

A year or so ago, I found some online discussion about this street-sign trouble, but now I can't locate the URLs. I did find the following related link, which illustrates related problems:

General problems with traffic signs –mostly in the Boston area

I liked this film quite a bit. You can read about it at IMDB.

This was one of the in-flight films on an Air France flight (Paris–>Boston) that I took recently. I'm not used to finding good films on airplanes, so I was quite surprised; it was enjoyable, thoughtful, and funny.

I note that there seems to have been a fairly large flame-fest in the "postings" at IMDB about this movie. So I suspect that some will not agree with my assessment.

I recently read several books etc. by Thomas Frank, including What's the matter with Kansas? How conservatives won the heart of America (Metropolitan Books, 2004) and One Market Under God: Extreme Capitalism, Market Populism and the End of Economic Democracy (Doubleday, New York, 2000) I highly recommend both. While I'm being an advocate, let me point to Frank's web site, where you will find a variety of politically interesting links etc.

I'll also point to the very nice journal The Baffler for which Frank is an editor.

Frank wrote in "Le Monde Diplomatique" a quite prescient piece nine months or so before the last US presidential election, which may be found HERE

My mountain bike has met with some unfortunate fate, so I bought a used bike which is a single-speed, and will make a better commute bike. It is built up on an old Univega road-frame. It is fun to ride!

I'm on my way to Eugene OR for the 60th Birthday conference for my Ph.D. advisor, Gary Seitz. Right now I'm sitting in the South Bend airport (Michiana Air-something-or-other). And there is a local "famous" person in the airport: congressman Chocola. It is very odd watching everyone walking up and chatting with (telling him "you have our vote" etc). Odd, I say, because I can't stand his politics: Yuck.

I loved this film! It is a documentary Tony Wilson and Factory Records, a Manchester UK record company that was formed during the "revolution" in popular music begun by the Sex Pistols in Britain. Joy Division and later New Order recorded for this label. I think this film fascinates me because I never understood as an adolescent how it was that bands with a punk origin started making what is essentially dance music; somehow this bugged the 18 year old version of me, but doesn't seem to bother me at all

I received a big pile of books for christmas, and was so excited I seem to have begun three of them more-or-less at once – I seem to be saving the fiction for later. I'm reading:

• Lipstick Traces by Greil Marcus
• Imagining Numbers by Barry Mazur

  and I just finished

• Mathematics and the roots of postmodern thought by Tasic.

It is worth pointing out that I was reminded to read Lipstick Traces by the (fun!) review of Tasic's book by Michael Harris in the Notices of the American Math Society; this review can be found here: Postmodern at an Early Age.

• Linear Algebraic Groups
• Front for the mathematics arXiv
• JSTOR
• NUMDAM Numérisation de documents anciens mathématiques
• Publications IHES at NUMDAM
• Goettinger DigitalisierungsZentrum (GDZ)

Here are some photos of the rather tragic demise of our car; we bought the Camry in 1996 – just as I finished graduate school in Eugene. It got squished by a tree last Monday night (Jul 7, 2003).

It is difficult to comment on the Iraq war without losing my temper. One of the early and striking pieces of insanity, though, was the demonization of France by the Bush administration.

Here is some discussion, taken from a 18 February 2003 Chicago tribune editorial by Justin Vaisse (visiting fellow at the Center on the U.S. and France at the Brookings Institution) called "Dissecting the French connection to war"

"How dare the French forget…" I apologize for being so ungrateful. It's just that I learned in school that France and Britain declared war on Nazi Germany in September 1939, while the United States was enacting isolationist laws, and that America entered the war two years later, only after Japan attacked Pearl Harbor. But now I see that was just Gallic propaganda. How could I have believed it?

Another thing I had failed to appreciate was how isolated we French are. It's painful to admit, but only 73 percent of the French people oppose a war without a second UN resolution. We definitely cannot pretend we speak for the rest of the world, as war is opposed by 82 percent of the European Union (84 percent of Brits), and in other parts of the world, let's say South America, it's more in the range of 90 percent. So we should shut up. And we should also admit that our isolation makes us insignificant, though I still can't understand why publications such as the Weekly Standard keep talking about us so much. Maybe it has something to do with our food.

My situation is now very difficult: When I talk to my former French friends on the phone, they claim they oppose the war for the same reasons about 40 percent of Americans do. They claim that they find their own arguments expounded in American newspapers by American statesmen; namely, that war would help Osama bin Laden recruit new followers; that war would trigger more terrorist attacks at home and abroad; that containment can work; and that it would be hard to impose stability – let alone democracy – on Iraq.