why nag? (3)
Here is
the construction of this normal space or chart
494ad6c5b11eb5acaeb1f04c713752e7.gif' title='\mathbf{chart}_{\Gamma}'
alt='\mathbf{chart}_{\Gamma}' />. The sub-semigroup of
6c26a35d57cea85ca5ba8a6906294f60.gif' title='Z^5' alt='Z^5' /> (all
dimension vectors of Q) consisting of those vectors
418dd64ba3cd4289441d53fe254876d3.gif'
title='\alpha=(a_1,a_2,b_1,b_2,b_3)' alt='\alpha=(a_1,a_2,b_1,b_2,b_3)'
/> satisfying the numerical condition
a644a1075ef65abc7cbbeb14846c5c99.gif' title='a_1+a_2=n=b_1+b_2+b_3'
alt='a_1+a_2=n=b_1+b_2+b_3' /> is generated by six dimension vectors,
namely those of the 6 non-isomorphic one-dimensional solutions in
e1656285d338b15de3df70e214fbd051.gif' title='\mathbf{rep}~\Gamma'
alt='\mathbf{rep}~\Gamma' />
a59961f7695f6a329d40347aa9f16498.gif' title='S_1 = \xymatrix@=.4cm{ & &
& & \vtx{1} \\ \vtx{1} \ar[rrrru]^1 \ar[rrrrd] \ar[rrrrddd] & & & & \\ &
& & & \vtx{0} \\ \vtx{0} \ar[rrrruuu] \ar[rrrru] \ar[rrrrd] & & & & \\ &
& & & \vtx{0}} \qquad S_2 = \xymatrix@=.4cm{ & & & & \vtx{0} \\ \vtx{0}
\ar[rrrru] \ar[rrrrd] \ar[rrrrddd] & & & & \\& & & & \vtx{1} \\\vtx{1}
\ar[rrrruuu] \ar[rrrru]^1 \ar[rrrrd] & & & & \\ & & & & \vtx{0}}'
alt='S_1 = \xymatrix@=.4cm{ & & & & \vtx{1} \\ \vtx{1} \ar[rrrru]^1
\ar[rrrrd] \ar[rrrrddd] & & & & \\ & & & & \vtx{0} \\ \vtx{0}
\ar[rrrruuu] \ar[rrrru] \ar[rrrrd] & & & & \\ & & & & \vtx{0}} \qquad
S_2 = \xymatrix@=.4cm{ & & & & \vtx{0} \\ \vtx{0} \ar[rrrru] \ar[rrrrd]
\ar[rrrrddd] & & & & \\& & & & \vtx{1} \\\vtx{1} \ar[rrrruuu]
\ar[rrrru]^1 \ar[rrrrd] & & & & \\ & & & & \vtx{0}}' />
2bf710496d35295d2ae9b7a3322d0fab.gif' title='S_3 = \xymatrix@=.4cm{ & &
& & \vtx{0} \\ \vtx{1} \ar[rrrru] \ar[rrrrd] \ar[rrrrddd]^1 & & & & \\ &
& & & \vtx{0} \\ \vtx{0} \ar[rrrruuu] \ar[rrrru] \ar[rrrrd] & & & & \\ &
& & & \vtx{1}} \qquad S_4 = \xymatrix@=.4cm{ & & & & \vtx{1} \\ \vtx{0}
\ar[rrrru] \ar[rrrrd] \ar[rrrrddd] & & & & \\ & & & & \vtx{0} \\ \vtx{1}
\ar[rrrruuu]^1 \ar[rrrru] \ar[rrrrd] & & & & \\ & & & & \vtx{0}}'
alt='S_3 = \xymatrix@=.4cm{ & & & & \vtx{0} \\ \vtx{1} \ar[rrrru]
\ar[rrrrd] \ar[rrrrddd]^1 & & & & \\ & & & & \vtx{0} \\ \vtx{0}
\ar[rrrruuu] \ar[rrrru] \ar[rrrrd] & & & & \\ & & & & \vtx{1}} \qquad
S_4 = \xymatrix@=.4cm{ & & & & \vtx{1} \\ \vtx{0} \ar[rrrru] \ar[rrrrd]
\ar[rrrrddd] & & & & \\ & & & & \vtx{0} \\ \vtx{1} \ar[rrrruuu]^1
\ar[rrrru] \ar[rrrrd] & & & & \\ & & & & \vtx{0}}' />
59fe70dac8703efdbb3f8abb19188d9e.gif' title='S_5 = \xymatrix@=.4cm{ & &
& & \vtx{0} \\ \vtx{1} \ar[rrrru] \ar[rrrrd]^1 \ar[rrrrddd] & & & & \\ &
& & & \vtx{1} \\ \vtx{0} \ar[rrrruuu] \ar[rrrru] \ar[rrrrd] & & & & \\ &
& & & \vtx{0}} \qquad S_6 = \xymatrix@=.4cm{ & & & & \vtx{0} \\ \vtx{0}
\ar[rrrru] \ar[rrrrd] \ar[rrrrddd] & & & & \\ & & & & \vtx{0} \\ \vtx{1}
\ar[rrrruuu] \ar[rrrru] \ar[rrrrd]^1 & & & & \\ & & & & \vtx{1}}'
alt='S_5 = \xymatrix@=.4cm{ & & & & \vtx{0} \\ \vtx{1} \ar[rrrru]
\ar[rrrrd]^1 \ar[rrrrddd] & & & & \\ & & & & \vtx{1} \\ \vtx{0}
\ar[rrrruuu] \ar[rrrru] \ar[rrrrd] & & & & \\ & & & & \vtx{0}} \qquad
S_6 = \xymatrix@=.4cm{ & & & & \vtx{0} \\ \vtx{0} \ar[rrrru] \ar[rrrrd]
\ar[rrrrddd] & & & & \\ & & & & \vtx{0} \\ \vtx{1} \ar[rrrruuu]
\ar[rrrru] \ar[rrrrd]^1 & & & & \\ & & & & \vtx{1}}' />
In
particular, in any component
50a80f203430b8fa8b746e21a6031866.gif' title='\mathbf{rep}_{\alpha}~Q'
alt='\mathbf{rep}_{\alpha}~Q' /> containing an open subset of
representations corresponding to solutions in
e1656285d338b15de3df70e214fbd051.gif' title='\mathbf{rep}~\Gamma'
alt='\mathbf{rep}~\Gamma' /> we have a particular semi-simple solution
073019e329f9c1a6ab20d34241a7cb2f.gif' title='M = S_1^{\oplus g_1} \oplus
S_2^{\oplus g_2} \oplus S_3^{\oplus g_3} \oplus S_4^{\oplus g_4} \oplus
S_5^{\oplus g_5} \oplus S_6^{\oplus g_6}' alt='M = S_1^{\oplus g_1}
\oplus S_2^{\oplus g_2} \oplus S_3^{\oplus g_3} \oplus S_4^{\oplus g_4}
\oplus S_5^{\oplus g_5} \oplus S_6^{\oplus g_6}' />
and in
particular
90177bc293e0ae558275d34fa1f7b402.gif' title='\alpha =
(g_1+g_3+g_5,g_2+g_4+g_6,g_1+g_4,g_2+g_5,g_3+g_6)' alt='\alpha =
(g_1+g_3+g_5,g_2+g_4+g_6,g_1+g_4,g_2+g_5,g_3+g_6)' />. The normal space
to the
57cdc94ad4d4036fc60f98d6b9a93914.gif' title='GL(\alpha)'
alt='GL(\alpha)' />-orbit of M in
50a80f203430b8fa8b746e21a6031866.gif' title='\mathbf{rep}_{\alpha}~Q'
alt='\mathbf{rep}_{\alpha}~Q' /> can be identified with the representation
space
857a7c502af2501c86c80d1d95bf0efa.gif' title='\mathbf{rep}_{\beta}~Q'
alt='\mathbf{rep}_{\beta}~Q' /> where
bd038a0e304d7d1b3ee2c5d71c1f9040.gif' title='\beta=(g_1,\ldots,g_6)'
alt='\beta=(g_1,\ldots,g_6)' /> and Q is the quiver of the following
form
ec5bfddac46a6eed7dbbd791918b9aed.gif' title='\xymatrix{ & \vtx{g_1}
\ar@/^/[ld]^{C_{16}} \ar@/^/[rd]^{C_{12}} & \\ \vtx{g_6}
\ar@/^/[ru]^{C_{61}} \ar@/^/[d]^{C_{65}} & & \vtx{g_2}
\ar@/^/[lu]^{C_{21}} \ar@/^/[d]^{C_{23}} \\ \vtx{g_5}
\ar@/^/[u]^{C_{56}} \ar@/^/[rd]^{C_{54}} & & \vtx{g_3}
\ar@/^/[u]^{C_{32}} \ar@/^/[ld]^{C_{34}} \\ & \vtx{g_4}
\ar@/^/[lu]^{C_{45}} \ar@/^/[ru]^{C_{43}} & }' alt='\xymatrix{ &
\vtx{g_1} \ar@/^/[ld]^{C_{16}} \ar@/^/[rd]^{C_{12}} & \\ \vtx{g_6}
\ar@/^/[ru]^{C_{61}} \ar@/^/[d]^{C_{65}} & & \vtx{g_2}
\ar@/^/[lu]^{C_{21}} \ar@/^/[d]^{C_{23}} \\ \vtx{g_5}
\ar@/^/[u]^{C_{56}} \ar@/^/[rd]^{C_{54}} & & \vtx{g_3}
\ar@/^/[u]^{C_{32}} \ar@/^/[ld]^{C_{34}} \\ & \vtx{g_4}
\ar@/^/[lu]^{C_{45}} \ar@/^/[ru]^{C_{43}} & }' />
and we can
even identify how the small matrices
4cf2e53cd44656e74461d8e8b3596a6a.gif' title='C_{ij}' alt='C_{ij}' /> fit
into the
fac654be75426dc302e993b74bfbdc4b.gif' title='3 \times 2' alt='3 \times
2' /> block-decomposition of the base-change matrix B
38560f69869a312524a988566f327ce6.gif' title='B = \begin{bmatrix}
\begin{array}{ccc|ccc} 1_{a_1} & 0 & 0 & C_{21} & 0 & C_{61} \\ 0 &
C_{34} & C_{54} & 0 & 1_{a_4} & 0 \\ \hline C_{12} & C_{32} & 0 &
1_{a_2} & 0 & 0 \\ 0 & 0 & 1_{a_5} & 0 & C_{45} & C_{65} \\ \hline 0 &
1_{a_3} & 0 & C_{23} & C_{43} & 0 \\ C_{16} & 0 & C_{56} & 0 & 0 &
1_{a_6} \\ \end{array} \end{bmatrix}' alt='B = \begin{bmatrix}
\begin{array}{ccc|ccc} 1_{a_1} & 0 & 0 & C_{21} & 0 & C_{61} \\ 0 &
C_{34} & C_{54} & 0 & 1_{a_4} & 0 \\ \hline C_{12} & C_{32} & 0 &
1_{a_2} & 0 & 0 \\ 0 & 0 & 1_{a_5} & 0 & C_{45} & C_{65} \\ \hline 0 &
1_{a_3} & 0 & C_{23} & C_{43} & 0 \\ C_{16} & 0 & C_{56} & 0 & 0 &
1_{a_6} \\ \end{array} \end{bmatrix}' />
Hence, it makes sense
to call Q the non-commutative normal space to the isomorphism problem in
e1656285d338b15de3df70e214fbd051.gif' title='\mathbf{rep}~\Gamma'
alt='\mathbf{rep}~\Gamma' />. 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
e1656285d338b15de3df70e214fbd051.gif' title='\mathbf{rep}~\Gamma'
alt='\mathbf{rep}~\Gamma' />.
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
bd038a0e304d7d1b3ee2c5d71c1f9040.gif' title='\beta=(g_1,\ldots,g_6)'
alt='\beta=(g_1,\ldots,g_6)' /> with
b79713c8c508ada1523aab174c69d630.gif' title='n = \gamma_1 + \ldots +
\gamma_6' alt='n = \gamma_1 + \ldots + \gamma_6' />, then for every
non-zero scalar
5dbac7ebb9d77dbe0017507cc30538ac.gif' title='\lambda \in \mathbb{C}^*'
alt='\lambda \in \mathbb{C}^*' /> the matrices
32bd2d1abbf682daa8f8023ab1503778.gif' title='X = \lambda B^{-1}
\begin{bmatrix} 1_{g_1+g_4} & 0 & 0 \\ 0 & \rho^2 1_{g_2+g_5} & 0 \\ 0 &
0 & \rho 1_{g_3+g_6} \end{bmatrix} B \begin{bmatrix} 1_{g_1+g_3+g_5} & 0
\\ 0 & -1_{g_2+g_4+g_6} \end{bmatrix}' alt='X = \lambda B^{-1}
\begin{bmatrix} 1_{g_1+g_4} & 0 & 0 \\ 0 & \rho^2 1_{g_2+g_5} & 0 \\ 0 &
0 & \rho 1_{g_3+g_6} \end{bmatrix} B \begin{bmatrix} 1_{g_1+g_3+g_5} & 0
\\ 0 & -1_{g_2+g_4+g_6} \end{bmatrix}' />
52b74cec40450fb2e4c3b6fda3c805e7.gif' title='Y = \lambda \begin{bmatrix}
1_{g_1+g_3+g_5} & 0 \\ 0 & -1_{g_2+g_4+g_6} \end{bmatrix} B^{-1}
\begin{bmatrix} 1_{g_1+g_4} & 0 & 0 \\ 0 & \rho^2 1_{g_2+g_5} & 0 \\ 0 &
0 & \rho 1_{g_3+g_6} \end{bmatrix} B' alt='Y = \lambda \begin{bmatrix}
1_{g_1+g_3+g_5} & 0 \\ 0 & -1_{g_2+g_4+g_6} \end{bmatrix} B^{-1}
\begin{bmatrix} 1_{g_1+g_4} & 0 & 0 \\ 0 & \rho^2 1_{g_2+g_5} & 0 \\ 0 &
0 & \rho 1_{g_3+g_6} \end{bmatrix} B' />
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
740c53ea0fbc3afa50a1b90cfa7ea2cf.gif' title='g_i \leq g_{i-1} + g_{i+1}'
alt='g_i \leq g_{i-1} + g_{i+1}' />
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
5b054112bf1f88f61b7d34e3e4ce0c7e.gif' title='\mathbf{rep}~B_3'
alt='\mathbf{rep}~B_3' /> 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
b0603860fcffe94e5b8eec59ed813421.gif' title='\beta' alt='\beta'
/>-dimensional representations of the quiver Q, then the corresponding
solutions
f00acce613318349cb04ab296486fc11.gif' title='(X,Y)' alt='(X,Y)' /> 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
b0603860fcffe94e5b8eec59ed813421.gif' title='\beta' alt='\beta' />.
Finally, our approach also indicates why the classification of
braid-relation solutions of size
0ad4dcd2bcf969a27df1a7a9f46947eb.gif' title='\leq 5' alt='\leq 5' /> is
easier : from size 6 on there are new classes of simple
Q-representations given by going round the whole six-cycle!