why nag? (1)

Let us take a hopeless problem, motivate why something like non-commutative algebraic geometry might help to solve it, and verify whether this promise is kept.

Suppose we want to know all solutions in invertible matrices to the braid relation (or Yang-Baxter equation)

All such solutions (for varying size of matrices) form an additive Abelian category , so a big step forward would be to know all its simple solutions (that is, those whose matrices cannot be brought in upper triangular block form). A literature check shows that even this task is far too ambitious. The best result to date is the classification due to Imre Tuba and Hans Wenzl of simple solutions of which the matrix size is at most 5.

For fixed matrix size n, finding solutions in is the same as solving a system of cubic polynomial relations in unknowns, which quickly becomes a daunting task. Algebraic geometry tells us that all solutions, say form an affine closed subvariety of -dimensional affine space. If we assume that is a smooth variety (that is, a manifold) and if we know one solution explicitly, then we can use the tangent space in this point to linearize the problem and to get at all solutions in a neighborhood.

So, here is an idea : assume that itself would be a non-commutative manifold, then we might linearize our problem by considering tangent spaces and obtain new solutions out of already known ones. But, what is a non-commutative manifold? Well, by the above we at least require that for all integers n the commutative variety is a commutative manifold.

But, there is still some redundancy in our problem : if is a solution, then so is any conjugated pair where is a basechange matrix. In categorical terms, we are only interested in isomorphism classes of solutions. Again, if we fix the size n of matrix-solutions, we consider the affine variety as a variety with a -action and we like to classify the orbits of simple solutions. If is a manifold then the theory of Luna slices provides a method, both to linearize the problem as well as to reduce its complexity. Instead of the tangent space we consider the normal space N to the -orbit (in a suitable solution). On this affine space, the stabilizer subgroup acts and there is a natural one-to-one correspondence between -orbits in and -orbits in the normal space N (at least in a neighborhood of the solution).

So, here is a refinement of the idea : we would like to view as a non-commutative manifold with a group action given by the notion of isomorphism. Then, in order to get new isoclasses of solutions from a constructed one we want to reduce the size of our problem by considering a linearization (the normal space to the orbit) and on it an easier isomorphism problem.

However, we immediately encounter a problem : calculating ranks of Jacobians we discover that already is not a smooth variety so there is not a chance in the world that might be a useful non-commutative manifold. Still, if is a solution to the braid relation, then the matrix commutes with both X and Y.

If is a simple solution, this means that after performing a basechange, becomes a scalar matrix, say . But then, is a solution to

and all such solutions form a non-commutative closed subvariety, say of and if we know all (isomorphism classes of) simple solutions in we have solved our problem as we just have to bring in the additional scalar .

Here we strike gold : is indeed a non-commutative manifold. This can be seen by identifying with one of the most famous discrete infinite groups in mathematics : the modular group . The modular group acts by Mobius transformations on the upper half plane and this action can be used to write as the free group product . Finally, using classical representation theory of finite groups it follows that indeed all are commutative manifolds (possibly having many connected components)! So, let us try to linearize this problem by looking at its non-commutative tangent space, if we can figure out what this might be.

Here is another idea (or rather a dogma) : in the world of non-commutative manifolds, the role of affine spaces is played by the representations of finite quivers Q. A quiver is just on oriented graph and a representation of it assigns to each vertex a finite dimensional vector space and to each arrow a linear map between the vertex-vector spaces. The notion of isomorphism in is of course induced by base change actions in all of these vertex-vector spaces. (to be continued)