M Theory Lesson 96
Non-commutative polynomials are built out of words in an alphabet of letters. The simplest case is two letters $X$ and $Y$. If we imagine, as pointed out in this helpful paper, that the letter $X$ represents $\frac{dx}{x}$ and the letter $Y$ represents $\frac{dx}{1 - x}$ then an MZV may be expressed as an iterated integral
$\zeta (s_1, s_2, \cdots , s_k) = \int_{0}^{1} X^{s_1 - 1} Y X^{s_2 - 1} Y \cdots X^{s_k - 1} Y$
in dimension $d = \sum s_i$. Given any alphabet whatsoever, a shuffle is the operation sending words $V$ and $W$ to the sum of permutations on letters which preserve the order of letters within the words. Thus MZVs obey a shuffle product.
A two letter alphabet also arises in the Lorenz template representation of knots, where words in the two holes describe a knot or link. In fact, MZV algebras were initially studied in relation to the homfly polynomial as formulated by Kontsevich.
$\zeta (s_1, s_2, \cdots , s_k) = \int_{0}^{1} X^{s_1 - 1} Y X^{s_2 - 1} Y \cdots X^{s_k - 1} Y$
in dimension $d = \sum s_i$. Given any alphabet whatsoever, a shuffle is the operation sending words $V$ and $W$ to the sum of permutations on letters which preserve the order of letters within the words. Thus MZVs obey a shuffle product.
A two letter alphabet also arises in the Lorenz template representation of knots, where words in the two holes describe a knot or link. In fact, MZV algebras were initially studied in relation to the homfly polynomial as formulated by Kontsevich.
posted by Kea | 2:37 PM
7 Comments:
So the dimension is allowed to be complex?
Hi Kea,
I have always wondered whether you write in non-processed latex for a reason or if it is my browser. Now that wordpress allows latex and processes it alright, I wonder more. Am I the only one to see untranslated $\frac ... scribblings in your posts ?
Cheers,
T.
Tommaso, I hope you are one of few people to see gobbledygook. Anyone who downloads a few mathML fonts should be able to see decent formatting, and most internet cafes I visit have browsers for which my site looks OK. I will move to wordpress at some point, hopefully.
Tommaso, have you tried viewing this site with the firefox web browser?
Good question, kneemo. Here the s_i are just positive integers, otherwise we wouldn't have polynomials. But in general one is interested in 'analytic continuations' of zeta functions. Since the positive integers come from a 1-operad index, I have been trying to think of more general zeta arguments in terms of higher operad trees, or surreal trees and generalisations thereof. I'm sorry if I didn't make that clear before. As we have seen, the MZV algebra is associated to the associahedra/ModuliStack integrals, and is therefore only representative of a low level (one value) of the q heirarchy. Matti and I are thinking that the full zeta functions must be understood in terms of all q.
One could understand some algebraic complex numbers as "supernaturals", p-adic numbers which are infinite as real numbers, as previous discussions about rigs demonstrated.
For instance, Golden Object of John Baez could be realized in this manner with -1 interpreted as supernatural (p-1)(1+p+p^2+-...). I wrote about this a posting to my own blog.
For a given prime p, a considerable subset for algebraic values of arguments of zeta could be understood as supernaturals.
Algebraic valued complex dimensions could be understood as super-naturals, and thus real but infinite, for an infinite set of primes. Infinite dimension could thus be infinite in very many manners. It would be interesting if one could assign this kind of dimension to various variants of say Hilbert space resulting as various infinite-D sub-spaces of Hilbert space.
Thanks, Matti, that's much clearer.
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