In this post, I want to talk about a really cool and surprising connection between simple Lie algebras (via Dynkin diagrams) and certain singularities of quotient varieties. More generally, this defines a correspondence between simple Lie algebras and some discrete subgroups of , which is a small part of Arnol’d’s “mathematical trinities.” At the end of the post, I include Arnol’d’s amazing table, with a link to his exposition of it. Thanks to Boris Alexeev and Tamar Friedman for introducing this material to me. For the most part, I follow online notes by Miles Reid. The singularities themselves (rather than Arnol’d’s table) are standard fair in algebraic surface theory, often under the heading of “Duval singularities.”
Let be complex affine 3-space, and let be the zero set of
It is a hypersurface (2-dimensional over ) with singularity at . Geometrically, its real part looks like two cones symmetric about the origin, joining at a singular point. The cones are a union of “generating lines” which all meet at . The “blow up'” of at is simply a cylinder, whose underlying set is the disjoint union of these lines; thus the point , in the real case, is replaced by a circle, called the exceptional divisor of the blow up. Over the complex numbers, the exceptional divisor is a copy of . The resulting curve is non-singular, so we have “resolved the singularity” at . During this post, the exceptional divisor will always turn out to be the union of ‘s which intersect in a finite number of points. In this case, the intersection diagram of the exceptional divisor is defined to be the planar graph whose points are the copies of , and whose edges correspond to the intersections. Therefore the diagram in the case of at is a single vertex.
As we’ve just seen, the idea of blowing up is to “pull apart” the multiple tangent lines at a singular point; if the resulting algebraic set is still singular, we may have to blow up more times to resolve the singularities. Here’s a more formal definition. There is a natural rational map which sends to and is defined everywhere but the origin. Thinking of as the space of lines through in , then is the line from to when . The blow up of will be essentially the graph of restricted to ; as parts of approach the singularity at along different lines, they will no longer collide in the graph, thus pulling apart the tangent lines (skip ahead to picture).
Formally, the graph of is defined to be the subspace of with two pieces: the first consisting of all elements which we call , and the second just (call it ). So we see that the graph projects isomorphically to everywhere away from the origin, just by restricting to the first coordinate, and above the origin there is a whole copy of . Then the blow up is defined as the Zariski closure of the first part (the Zariski closure of a subset is just the smallest algebraic set containing ). The subset of which projects to the origin in is called the exceptional divisor; it is exactly .
Here are two pictures to clarify this, both stolen shamelessly from Harris’s introduction (a great book!). The first shows the blow up of , (over the reals). From the picture one can see that every point downstairs corresponds to one point in the blow up, and points in the same line through downstairs are at the same height upstairs. The line over the origin is a copy of . The second picture is the blow up of the nodal cubic, which is just a subset of the blow up of . One can see how the two different tangent directions at are pulled apart into a non-singular curve.
Given , an element is equal to if and only if for all . So another definition of the graph of restricted to (i.e. ) is as the set
Using this description, we can actually compute the blow up of some surfaces. For example, let’s go back to
Suppose we write an element of as . Then points of the blow up satisfy
Restricting, say, to corresponds to restricting to an affine chart in . In this case we combine the above equations to get
The first irreducible piece corresponds to , which is a copy of lying above (as it should be). Taking the Zariski closure of the graph minus thus corresponds to restricting to the other piece , which is a smooth surface in (and one can check that the blow up is smooth in the other charts as well). Remaining in the chart, restricting to gives the exceptional divisor, which is a curve in , and by degree arguments (e.g. it’s a hyperelliptic curve of degree 1 in , so its genus is 0 by the Riemann Hurwitz theorem) it must be the affine part of a copy of , as we claimed before.
The surface also has another realization as a quotient variety. Namely, the group acts on via , and the quotient of by this action gives a space which is a non-singular algebraic set everywhere except at the origin, where it is singular. The map sending descends to this quotient variety, defining an isomorphism between the quotient variety and .
In this way, any discrete subgroup of whose only fixed point is the origin will define a quotient variety which is non-singular except possibly at . For example, the binary dihedral group of order 8, which is a lift of a dihedral group to , defines an action on generated by the two maps
Then the map from sending
presents this quotient variety, which we call , as a hypersurface in ; we can change coordinates so that the hypersurface is the zero set of
Now, let’s blow up the singularity at for as we did for . As before, we take the chart and get the equations
Substituting, we obtain
so the blow up in this chart is just the (singular) surface
after we discard . The exceptional divisor corresponds to , whence , so it is just the axis, giving a copy of . Call this . In the other charts, we get
From these equations we can see that all singularities were present in the first chart, at points . At each of these points, we can change coordinates so that the singularity looks like the singularity in . So we just have to blow up each of these points, and we will get something smooth. There is evidently a symmetry of the surface interchanging them, so, to compute the entire intersection diagram, it suffices to consider what happens at any one of them.
We take a fresh set of coordinates , and blow up the curve
at some singularity. To do this, take , and define ; then our equation becomes
which is singular at . Blowing up as before and again taking the chart, we get
The exceptional divisor is the intersection of this surface with the plane , which is , again a copy of . Blowing up in this chart, we get the axis, i.e., . So the intersection of the blow up of with is the point . By symmetry, the other exceptional divisors (corresponding to the other third roots of -1) intersect in the same way, so our final intersection diagram is exactly
with in the middle. This is the Dynkin diagram corresponding to the lie algebra ! Of course, the intersection diagram for was the dynkin diagram . Thus we have a correspondance of groups and diagrams, , . In fact, this completely generalizes. Here is Reid’s table of all the Duval singularities:
And here, as promised, is Arnol’d’s:
Play around with this for a while! The table has an exposition in Arnol’d’s book, here (.djvu)
Tamar has generalized this up a dimension or two; she found that the same thing works as long as you DEFINE AN ENTIRELY NEW LIE ALGEBRA-LIKE STRUCTURE! She calls them Lie algebras of the third kind (LATKe’s). Here’s her paper.