Pure and Applied Mathematics Quarterly

Volume 19 (2023)

Number 1

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

DOI: https://dx.doi.org/10.4310/PAMQ.2023.v19.n1.a7


Noemie Combe (Max Planck Institut for Mathematics in Sciences, Leipzig, Germany)

Yuri I. Manin (Max Planck Institut for Mathematics, Bonn, Germany)

Matilde Marcolli (Depts. of Mathematics and of Computing and Mathematical Sciences, California Institute of Technology, Pasadena, Calif., U.S.A.)


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