Altmetric

A structure theorem for streamed information

File Description SizeFormat 
1-s2.0-S0021869323003721-main.pdfPublished version516.15 kBAdobe PDFView/Open
Title: A structure theorem for streamed information
Authors: Salvi, C
Diehl, J
Lyons, T
Preiss, R
Reizenstein, J
Item Type: Journal Article
Abstract: We identify the free half shuffle algebra of Schützenberger [31] with an algebra of real-valued functionals on paths, where the half shuffle emulates the integration of a functional against another. We then provide two, to our knowledge, new identities in arity 3 involving its commutator (area), and show that these are sufficient to recover the Zinbiel and Tortkara identities introduced by Dzhumadil'daev [11]. We then use these identities to provide a simple proof of the main result of Diehl et al. [8], namely that any element of the free half shuffle algebra can be expressed as a polynomial over iterated areas. Moreover, we consider minimal sets of Hall iterated integrals defined through the recursive application of the half shuffle product to Hall trees. Leveraging the duality between this set of Hall integrals and classical Hall bases of the free Lie algebra, we prove using combinatorial arguments that any element of the free half shuffle algebra can be written uniquely as a polynomial over Hall integrals. We interpret this result as a structure theorem for streamed information, loosely analogous to the unique prime factorisation of integers, allowing to split any real valued function on streamed data into two parts: a first that extracts and packages the streamed information into recursively defined atomic objects (Hall integrals), and a second that evaluates a polynomial function in these objects without further reference to the original stream. The question of whether a similar result holds if Hall integrals are replaced by Hall areas is left as an open conjecture. Finally, we construct a canonical, but to our knowledge, new decomposition of the free half shuffle algebra as shuffle power series in the greatest letter of the original alphabet with coefficients in a sub-algebra freely generated by a new alphabet with an infinite number of letters. We use this construction to provide a second proof of our structure theorem.
Issue Date: 15-Nov-2023
Date of Acceptance: 1-Aug-2023
URI: http://hdl.handle.net/10044/1/112119
DOI: 10.1016/j.jalgebra.2023.07.024
ISSN: 0021-8693
Publisher: Elsevier
Start Page: 911
End Page: 938
Journal / Book Title: Journal of Algebra
Volume: 634
Copyright Statement: © 2023 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
Publication Status: Published
Online Publication Date: 2023-08-03
Appears in Collections:Financial Mathematics
Mathematics



This item is licensed under a Creative Commons License Creative Commons