The Jordan algebra rule
$(xx)(yx) = ((xx)y)x$
may be written in a multilinear form with six terms:
$(xy)(tz) + (zx)(ty) + (yz)(tx) = ((xy)t)z + ((zx)t)y + ((yz)t)x$
Observe that in this rule $x$, $y$ and $z$ exhibit the cyclic symmetry of a Jacobi rule. Setting $z = 1$ we have the reduced rule
$(xy)t + x(ty) + y(tx) = (xy)t + (xt)y + (yt)x$
which resembles two copies of a Jacobi rule. That is, since the Jordan algebra product is commutative, we can rewrite this expression as the trivial identity
$t(xy) + x(yt) + y(tx) = (xy)t + (yt)x + (tx)y$
so that the difference between the left and right hand Jacobi rules is only the swapping of bracketed terms. Allowing now for noncommutative elements, we can draw a general braided Jordan rule with six terms. In the diagram above, we ignore the crossing information.
Observe that the three terms on one side of the Jordan rule are all distinct non planar versions of the same rooted tree diagram. In other words, instead of one basic element of the pentagon associahedron, there are three copies. We may similarly draw non planar versions of any other rooted tree.
7 years ago