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.
Long term readers will recall the aim to use higher dimensional category theory (via physical motives and Batanin's polytopes) to generalise the case of the associahedra planar rooted trees. The associahedra are used to tile the (real points of the) moduli spaces for Riemann surfaces with $n$ marked points. Here we start to see 'cyclic trees', where we get $p$ copies of something where $p = 3$ will show up a lot in association with color and generation. See comments in the next post.
ReplyDeleteOK, so any Lie algebra (for instance) has a secret ternary structure, because of the Jacobi identity. We are always thinking in abstract terms: pre algebras, pre spacetimes. The cyclicity number $p$ is many things, such as the cardinality of the Fun Set, where Fun is that old field with one element, or dimension of MUB space, or ...
ReplyDeleteOne day some years ago (around 2006) I was sitting in a pub by Michael Batanin, and he was musing about cyclic generalisations of categorical sources and targets, somewhat along these motivic lines ... at the time, my ears pricked up, because I knew it somehow had to be the same thing that us dumb physicists were after, but it is taking a while for it to be worked out ...