User petra schwer - MathOverflow

Are certain simple Lie groups linear algebraic groups?

2012-09-19T15:01:08Z

Assume you have an almost connected simple Lie group G with trivial center. (In particular excluding non-algebraic examples such as the universal cover of SL_2(R).)

Such a group should automatically be an algebraic group over the reals resp. the complex numbers.

Is this true and why?

Can we in addition conclude (EDIT: under a good choice of the field and possibly additional assumptions?) that G is absolutely almost simple as an algebraic group?

EDIT: Asking this I do not want to regard a complex Lie group as a real algebraic group.

Answer by Petra Schwer for Terminology request: Graphs based on vector spaces

2012-09-17T08:50:20Z

Your situation reminds me of vertex colored graphs, that is graphs whose vertices are colored such that no two vertices of the same color share an edge. (See for example here: http://en.wikipedia.org/wiki/Graph_coloring ) You could then just use the dimension of the vector space associated to a vertex as its color. But it's not clear to me whether you objects satisfy this additional assumption on adjacency.

Answer by Petra Schwer for Justifying/Explaining math research in a public address

2012-08-08T11:59:22Z

I once attended a talk in which nothing but the classification of Platonic solids was explained to a general audience in order to show what math is like.

The speaker was doing really great in drawing beautiful pictures at the blackboard. Very little notation was used and no slides where involved. Even if you didn't manage to follow his arguments to the very end you could read off his fascination of his face and look at the pretty pictures.

He gave the talk in such a way that not a finished proof was presented but the object where explored together with the audience and steps of the proof where found in a live-discussion.

Maybe you can tell from what I am writing that this talk was really impressive and reached it's goal to teach that math can be a beautiful and fascinating way of exploring structures (or solving puzzles if you wish to say so).

Answer by Petra Schwer for Is this the CAT(0) metric on an affine building?

2011-08-29T17:23:15Z

The important thing seems to be that one needs to understand the connection between the CAT(0)-realization of the Coxeter complex that corresponds to the Weyl group and the description of an apartment given in terms of lattices. Since a geometric realization of this Coxeter complex will basically describe a geometric realization of one of the apartments in the building. By theorem 11.16 in the book by Abramenko and Brown (see Stefan Witzel's comment) the euclidean metric on one apartment determines the CAT(0)-metric on the entire building. So it is enough to understand the metric on one apartment.

Apartments correspond to ordered bases and, as you've said, the fundamental domain of the Weyl group action is the chamber (i.e. maximal simplex) with vertices $[[e_1, ..., e_i, te_{i+1}, ..., t e_n]]$, let's call it $c_0$.

One can realize the Weyl group as a euclidean reflection group. See Abramenko-Brown chapter 2.5.1. for a construction of the canonical linear representation. In addition, the corollary to proposition 5 in Bourbaki (Lie groups and Lie-algebras chapters 4-6) VI §2 chapter 2 gives explicit coordinates of the vertices of the fundamental chamber (called 'alcove' in Bourbaki, a 'chamber' is something else there) in terms of root systems. I.e. the chamber is described as a certain subset/simplex in some $R^n$.

In order to obtain the "standard-CAT(0)-metric" on the apartment defined by ${e_1, ... e_n}$ equip the simplex $c_0$ with the metric the fundamental chamber given in Bourbaki has. It should then be possible to check whether this is the same metric you've suggested above.

The following might be helpful, too: Davis, "Buildings are CAT(0)" http://www.math.osu.edu/~mdavis/buildings.pdf

Hope this makes sense and helps.

Answer by Petra Schwer for Longest element of Weyl groups.

2011-02-09T21:57:27Z

A good reference to try is Bourbaki "Lie groups and Lie Algebras, Chapters 4-6" Look at the plates at the end of the book, which contain all kinds of useful information about each one of the types.

Answer by Petra Schwer for Looking for figure of part of an A2 affine building

2011-02-08T19:50:43Z

A picture similar to the one you've made can be found in Garrett's book on Buildings and classical groups.

http://www.amazon.com/Buildings-classical-groups-Paul-Garrett/dp/041206331X

Comment by Petra Schwer

2012-09-19T16:09:52Z

Thanks for the reference! I'll have a look at it. Do you know how to make a precise statement for the real case?

Comment by Petra Schwer

2012-09-19T16:02:58Z

@Wildberd: So you say that the additional conclusion does not hold, yes?

Comment by Petra Schwer

2012-09-19T15:58:28Z

@Yves Cornulier: can we resolve this problem in general by passing to a finite cover of the group under consideration? Hence maybe editing the question such that we ask for some finite cover of the Lie group to be always algebraic?

Comment by Petra Schwer

2012-09-19T08:02:26Z

well, then sorry for the not really helpful remark.

Comment by Petra Schwer

2012-08-23T07:08:41Z

I just want to share this link to the New York Times with you. nytimes.com/2012/08/23/us/…

Comment by Petra Schwer

2012-08-15T12:03:18Z

Also, some Sudokus allow for several solutions which means the cues suffice to fill the grid but the solution is not unique. To find a physical solver you would then either need to assume uniqueness of the solution given the cues (but how does one test thi in advance??) or the solver should be able to give you all the solutions.

Comment by Petra Schwer

2011-02-08T19:46:28Z

yes - picked the wrong post. Sorry.