Here is
the construction of this normal space or chart . The sub-semigroup of (all
dimension vectors of Q) consisting of those vectors satisfying the numerical condition is generated by six dimension vectors,
namely those of the 6 non-isomorphic one-dimensional solutions in

In
particular, in any component containing an open subset of
representations corresponding to solutions in we have a particular semi-simple solution

and in
particular . The normal space
to the -orbit of M in can be identified with the representation
space where and Q is the quiver of the following
form

and we can
even identify how the small matrices fit
into the block-decomposition of the base-change matrix B

Hence, it makes sense
to call Q the non-commutative normal space to the isomorphism problem in
. Moreover, under this correspondence simple
representations of Q (for which both the dimension vectors and
distinguishing characters are known explicitly) correspond to simple
solutions in .

Having completed our promised
approach via non-commutative geometry to the classification problem of
solutions to the braid relation, it is time to collect what we have
learned. Let with , then for every
non-zero scalar the matrices

give a solution of size
n to the braid relation. Moreover, such a solution can be simple only if
the following numerical relations are satisfied

where indices are viewed
modulo 6. In fact, if these conditions are satisfied then a sufficiently
general representation of Q does determine a simple solution in and conversely, any sufficiently general simple n
size solution of the braid relation can be conjugated to one of the
above form. Here, by sufficiently general we mean a Zariski open (hence
dense) subset.

That is, for all integers n we have constructed
nearly all (meaning a dense subset) simple solutions to the braid
relation. As to the classification problem, if we have representants of
simple -dimensional representations of the quiver Q, then the corresponding
solutions of
the braid relation represent different orbits (up to finite overlap
coming from the fact that our linearizations only give an analytic
isomorphism, or in algebraic terms, an etale map). Such representants
can be constructed for low dimensional .
Finally, our approach also indicates why the classification of
braid-relation solutions of size is
easier : from size 6 on there are new classes of simple
Q-representations given by going round the whole six-cycle!

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