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Conway topograph, PGL(2)(Z)-dynamics and two-valued groups
journal contribution
posted on 2019-09-30, 10:03 authored by VM Buchstaber, Alexander VeselovAlexander VeselovConway’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 SurveysVolume
74Issue
3Pages
387 - 430Publisher
Turpion LTDVersion
- VoR (Version of Record)
Rights holder
© 2019 Russian Academy of Sciences (DoM), London Mathematical Society, Turpion Ltd.Publication date
2019-06Copyright date
2019ISSN
0036-0279eISSN
1468-4829Publisher version
Language
- en
Depositor
Prof Alexander VeselovUsage metrics
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Conway topographmodular grouptwo-valued groupsalgebraic discrete-time dynamicsintegrabilityScience & TechnologyPhysical SciencesMathematicsMODULAR GROUP ACTIONRATIONAL-POINTSMARKOV NUMBERSSURFACESALGEBRASDYNAMICSEQUATIONGROWTHFORMSGeneral MathematicsPure MathematicsApplied MathematicsMathematical Physics
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