Fricke Lie algebras and the genus zero property in Moonshine

Carnahan, Scott

doi:10.1088/1751-8121/aa781d

Cited by 4 publications

(1 citation statement)

References 33 publications

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“…Consequently, the spaces may be recognised 13 as defining a bosonic string theory in 26 dimensions. The generalised moonshine conjectures were recently proven 14 , 15 by Carnahan, so moonshine illuminates a physical origin for the monster, and for the 19 other sporadic groups that are involved in the monster. Therefore, 20 of the sporadic groups do indeed occur in nature.…”

Section: Introductionmentioning

confidence: 92%

Pariah moonshine

Duncan¹,

Mertens²,

Ono³

2017

Nat Commun

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Finite simple groups are the building blocks of finite symmetry. The effort to classify them precipitated the discovery of new examples, including the monster, and six pariah groups which do not belong to any of the natural families, and are not involved in the monster. It also precipitated monstrous moonshine, which is an appearance of monster symmetry in number theory that catalysed developments in mathematics and physics. Forty years ago the pioneers of moonshine asked if there is anything similar for pariahs. Here we report on a solution to this problem that reveals the O’Nan pariah group as a source of hidden symmetry in quadratic forms and elliptic curves. Using this we prove congruences for class numbers, and Selmer groups and Tate–Shafarevich groups of elliptic curves. This demonstrates that pariah groups play a role in some of the deepest problems in mathematics, and represents an appearance of pariah groups in nature.

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Section: Introductionmentioning

confidence: 92%

Pariah moonshine

Duncan¹,

Mertens²,

Ono³

2017

Nat Commun

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A Short Introduction to the Algebra, Geometry, Number Theory and Physics of Moonshine

Duncan

2020

Moscow Lectures

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Constructing a Lie group analog for the Monster Lie algebra

2022

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The Monster Lie algebra m, which admits an action of the Monster finite simple group M, was introduced by Borcherds as part of his work on the Conway-Norton Monstrous Moonshine conjecture. Here we construct an analog Gpmq of a Lie group, or Kac-Moody group, associated to m. The group Gpmq is given by generators and relations, analogous to the Tits construction of a Kac-Moody group. In the absence of local nilpotence of the adjoint representation of m, we introduce the notion of pro-summability of an infinite sum of operators. We use this to construct a complete pro-unipotent group p U `of automorphisms of a completion p m " n ´' h ' p n `of m, where p n `is the formal product of the positive root spaces of m. The elements of p U `are pro-summable infinite series with constant term 1. The group p U `has a subgroup p U ìm , which is an analog of a complete unipotent group corresponding to the positive imaginary roots of m. We construct analogs Exp : p n `Ñ p U `and Ad : p U `Ñ Autpp n `q of the classical exponential map and adjoint representation. Although the group Gpmq is not a group of automorphisms, it contains the analog of a unipotent subgroup U `, which conjecturally acts as automorphisms of p m. We also construct groups of automorphisms of m, of certain gl 2 subalgebras of m, of the completion p m and of similar completions of m that are conjecturally identified with subgroups of Gpmq.

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Fricke Lie algebras and the genus zero property in Moonshine

Cited by 4 publications

References 33 publications

Pariah moonshine

Pariah moonshine

A Short Introduction to the Algebra, Geometry, Number Theory and Physics of Moonshine

Constructing a Lie group analog for the Monster Lie algebra

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