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Conway topograph, PGL(2)(Z)-dynamics and two-valued groups

journal contribution
posted on 30.09.2019, 10:03 by VM Buchstaber, Alexander VeselovAlexander Veselov
Conway’s topographic approach to the binary quadratic forms and Markov triples is reviewed from the point of view of the theory of two-valued groups. This naturally leads to a new class of commutative two-valued groups, which we call involutive. We show that the two-valued group of Conway’s lax vectors plays a special role in this class. The group P GL2(Z), describing the symmetries of Conway topograph, acts by the automorphisms of this two-valued group. The binary quadratic forms are interpreted as primitive elements of Hopf 2-algebra of functions on the Conway group. This fact is used to construct an explicit embedding of the Conway two-valued group into R and thus to introduce the total group ordering on it. We classify all two-valued algebraic involutive groups with symmetric multiplication law and show that they are given by the coset construction from the addition law on the elliptic curves. In particular, this explains a special role of Mordell’s modification of the Markov equation and reveals its relation to the two-valued group from K-theory. We finish with the discussion of the role of two-valued groups and P GL2(Z) in the context of integrability in multi-valued dynamics.

History

School

  • Science

Department

  • Mathematical Sciences

Published in

Russian Mathematical Surveys

Volume

74

Issue

3

Pages

387 - 430

Publisher

Turpion LTD

Version

VoR (Version of Record)

Rights holder

© 2019 Russian Academy of Sciences (DoM), London Mathematical Society, Turpion Ltd.

Publication date

2019-06

Copyright date

2019

ISSN

0036-0279

eISSN

1468-4829

Language

en

Depositor

Prof Alexander Veselov