Pure and Applied Mathematics Quarterly
Volume 19 (2023)
Special Issue in honor of Don Zagier
Guest editors: Benedict H. Gross, Ken Ono, and Fernando Rodriguez Villegas
Moufang patterns and geometry of information
Pages: 149 – 189
Technology of data collection and information transmission is based on various mathematical models of encoding. The words “Geometry of information” refer to such models, whereas the words “Moufang patterns” refer to various sophisticated symmetries appearing naturally in such models.
In this paper, we show that the symmetries of spaces of probability distributions, endowed with their canonical Riemannian metric of information geometry, have the structure of a commutative Moufang loop. We also show that the $F$-manifold structure on the space of probability distribution can be described in terms of differential $3$-webs and Malcev algebras. We then present a new construction of (non-commutative) Moufang loops associated to almost-symplectic structures over finite fields, and use them to construct a new class of code loops with associated quantum error-correcting codes and networks of perfect tensors.
probability distributions, convex cones, Moufang loops, quasigroups, Malcev algebras, error-correcting codes, asymptotic bound, code loops, perfect tensors, tensor networks, CRSS quantum codes
2010 Mathematics Subject Classification
Primary 94-xx. Secondary 20Nxx, 46L07, 60E05.
N. C. Combe acknowledges support from the Minerva Fast track grant from the Max Planck Institute for Mathematics in the Sciences, in Leipzig.
Y. I. Manin acknowledges the continuing strong support from the Max Planck Institute for Mathematics in Bonn.
M. Marcolli acknowledges support from NSF grants DMS–1707882 and DMS-2104330.
Received 16 July 2021
Received revised 2 March 2022
Accepted 12 August 2022
Published 3 April 2023