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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