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John Baez (May 18 2020 at 21:14):Hi, @Joscha Diehl! What are you doing with nonassociative algebras? I have a fondness for those, and they're a fairly exotic topic.
(=_=) (May 19 2020 at 03:33):Joscha Diehl said:
Hi everyone, I am Joscha Diehl from Greifswald (https://diehlj.github.io/). Coming from Stochastic Analysis I'm interested in (noncommutative) algebras since they serve as book-keeping devices for various things there (the prime example being iterated integrals). This year I am looking into operads and homological algebra, to get a broader view on things. Happy to discuss noncommutative, nonassociative algebras, operads, as well as TDA!
Hi Joscha! How are iterated integrals used as "book-keeping devices" in stochastic analysis?
Joscha Diehl (May 19 2020 at 08:14):Hi @John Baez , thanks for the interest! Currently, one object of interest to me is the half-shuffle, also known as commutative dendriform algebra, or Zinbiel algebra ('Zinbiel' b/c of Koszul duality to Leibniz algebras). You might know this product from "Eilenberg S, MacLane S. 1953 On the groups H(π, n)."
Denote this product by a > b, then: a > (b >c) = (a>b + b>a) > c. I am interested in its antisymmetrization 'area(a,b) := a>b - b>a' (the half-shuffle corresponds to integration of smooth functions against each other, 'area' corresponds to calculating the area between them). It is a non-associative product; apart from anti-symmetry there is no other relation with two inputs; there is no new relation with three inputs; there is exactly one new relation with four inputs, sometimes called 'tortkara', which is believed to be the only additional relation.
The symmetrization of the half-shuffle, a>b+b>a, is the (associative and commutative !) shuffle product. Being allowed to apply the shuffle and 'area' product indiscriminately (think: binary rooted trees with 'shuffle' or 'area' decorating the inner nodes) one 'gets everything' (i.e. starting with generators of the half-shuffle algebra, which are letters of some finite alphabet, one generates the entire half-shuffle algebra).
For several reasons, we are interested in reducing this to (sums of) trees, where shuffles are only allowed to appear as a subtree connected to the root. In words: "One can apply 'area' to the generators as often as one desires, and with the results, one is only allowed to shuffle (but never to take areas again)."
We have an indirect proof that this works, but we don't really understand 'area' well enough yet, to have a direct proof.
Joscha Diehl (May 19 2020 at 08:20):Rongmin Lu said:
Joscha Diehl said:
Hi everyone, I am Joscha Diehl from Greifswald (https://diehlj.github.io/). Coming from Stochastic Analysis I'm interested in (noncommutative) algebras since they serve as book-keeping devices for various things there (the prime example being iterated integrals). This year I am looking into operads and homological algebra, to get a broader view on things. Happy to discuss noncommutative, nonassociative algebras, operads, as well as TDA!
Hi Joscha! How are iterated integrals used as "book-keeping devices" in stochastic analysis?
Hi @Rongmin Lu , thanks for asking. I was a bit sloppy with language here. The book-keeping devices are the algebras. Concretely, for iterated-integrals of a stochastic process (which are important for several reasons: numerics, asymptotics, statistics, control theory, rough path analysis, ...) one usually stores them as a multiplicative character on the "shuffle Hopf algebra". The product in this Hopf algebra is the (commutative, associative) shuffle product. This is nicely compatible with taking real-valued products of real-valued iterated-integrals. The coproduct is the (noncommutative, coassociative) deconcatenation coproduct, which is nicely compatible with the concatenation (as paths) of stochastic processes.
(=_=) (May 19 2020 at 08:28):@Joscha Diehl said:
Currently, one object of interest to me is the half-shuffle, also known as commutative dendriform algebra, or Zinbiel algebra ('Zinbiel' b/c of Koszul duality to Leibniz algebras). You might know this product from "Eilenberg S, MacLane S. 1953 On the groups H(π, n)."
I spent some time looking at Chen's iterated integrals back in my PhD days, trying to understand its use in cyclic homology. Zinbiel algebras were introduced by Loday and you should definitely check out his works.
Joscha Diehl (May 19 2020 at 08:32):Rongmin Lu
Parsing the works of Chen is still on my TODO list .. Did you every use it for anything?
Yes, Loday indeed has some nice conceptual works on Zinbiel/dendriform algebras.
(=_=) (May 19 2020 at 08:35):Joscha Diehl said:
Parsing the works of Chen is still on my TODO list .. Did you ever use it for anything?
Sadly, no. But check out the results for "iterated integrals" on Google Scholar and this nLab page.
(=_=) (May 19 2020 at 14:20):Joscha Diehl said:
For several reasons, we are interested in reducing this to (sums of) trees, where shuffles are only allowed to appear as a subtree connected to the root.
This is a very wild guess, and apologies if you're already aware of this, but you may or may not find #MIT Categories Seminar > April 16 - Joachim Kock's talk (the paper he's talking about is here) useful.