On Wednesday of this past week (1/29/2014) I went to the weekly colloquium in my department. The title was Involutions and “Fixing” the World: Symmetric Spaces and it was given by Catherine Buell of Bates College. The talk was interesting and I understood more than I thought I would (though I wouldn’t be able to say too much about it after the fact). What caught my attention was the mention of a maximum torus of a group.
Torus in a group? A subgroup that is a torus? In what way? There was talk of diagonal matrices, abelian subgroups etc. I really couldn’t follow that bit and I wasn’t the only person to ask about it. It’s particularly difficult to conceive of a subgroup being analogous or similar to a torus when the group is discrete such as the dihedral groups. Dr. Buell also studies semigroups and monoids for that matter.
What do you think of when I say “torus?”
A doughnut perhaps. But why? This is just a particular embedding of the torus in euclidean space. When I say torus why don’t you imagine,
Or for that matter why not the following, which is a fine embedding in 4-space. It’s only our limited perspective from 3-space that causes the self-intersection
So in what way are these tori? Are they the same? In what ways are they different? Barry Mazur has a great expository piece which is part category theory and part philosophy entitled “When is one thing equal to some other thing.” It’s available as a pdf on his website. I’ve only read it through once, so I probably should read it again, but I’d say that the point of the article is to convince you to step away from strict equality and move towards equality via isomorphism (and Mazur means that in a categorical sense, so an isomorphism in a category which is more general than the group isomorphisms you may be familiar with).
What’s a torus? I first met these objects in topology as I think this article will end up posing a number of questions and let’s get started. I suspect that usually we think of with a subspace topology inherited from the plane. Then the torus gets the resulting product topology.
Question 1) How does the topology of the circle affect the topology of the torus, are there any interesting tori, topologically?
If we think of the circle as a subset of the complex plane then we can associate it to the group U(1).
Question 2) What is the relationship between U(1) and ? Is a representation?
There’s certainly an isomorphism between U(1) and as a set of complex numbers (it’s certainly a group on it’s own right). As such the product groups are isomorphic as well, . So the torus can be thought of as an abelian group (the product of two abelian groups is abelian). Moreover, with the usual topology it’s connected and compact.
So the torus can be thought of, isomorphically, a compact connected abelian group. Actually we can say a bit more since it’s a manifold and a group so its a Lie group (provided it’s group operation and inverse are smooth maps).
If you Google ‘maximal torus’ you’ll find plenty. Elie Cartan has a theorem about them and some things called Weyl groups that I don’t understand yet. Here’s the definition of a maximal torus:
Definition: A maximal torus is a subgroup which is a torus and is maximal among all tori in G.
I’m really curious how many different sorts of groups we could talk about. It makes sense to me that Lie groups might have tori in them since they’re continuous, locally euclidean. There seems to be enough room, enough points to put a torus together. But how many groups have this property?
Proposition: A maximal torus is a maximal abelian subgroup
The converse is not true (they give an example but I don’t understand it fully and so will wait. It has to do with connectedness).
Question 3) Consider the set of diagonal 3×3 matrices with real entries. This forms an abelian subgroup of GL. But it’s not a torus, I don’t think, because it’s not compact. This set is homeomorphic to 3-space. What about the 3×3 diagonal matrices with determinant 1. Then they’ll be a subgroup of SO(3). Again, I think compactness is the problem as the values of the matrix will lie on the surface, .
Question 4) SO(n) is compact and connected and so will have a maximal torus subgroup. What is it?
Question 5) SL(n) is compact and connected and so will have a maximal torus subgroup. What is it?
Question 6) Same for SU(n)