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)

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