An infinite-dimensional 2-generated primitive axial algebra of Monster type

Clara Franchi; Mario Mainardis; Sergey Shpectorov

doi:10.1007/s10231-021-01157-8

An infinite-dimensional 2-generated primitive axial algebra of Monster type

Clara Franchi
, Mario Mainardis^*
, Sergey Shpectorov

^*Corresponding author for this work

Mathematics

Research output: Contribution to journal › Article › peer-review

2 Citations (Scopus)

89 Downloads (Pure)

Abstract

Rehren proved in Axial algebras. Ph.D. thesis, University of Birmingham (2015), Trans Am Math Soc 369:6953–6986 (2017) that a primitive 2-generated axial algebra of Monster type (α, β) , over a field of characteristic other than 2, has dimension at most 8 if α∉ { 2 β, 4 β}. In this note, we show that Rehren’s bound does not hold in the case α= 4 β by providing an example (essentially the unique one) of an infinite-dimensional 2-generated primitive axial algebra of Monster type (2,12) over an arbitrary field F of characteristic other than 2 and 3. We further determine its group of automorphisms and describe some of its relevant features.

Original language	English
Pages (from-to)	1279-1293
Number of pages	15
Journal	Annali di Matematica Pura ed Applicata
Volume	201
Early online date	22 Sept 2021
DOIs	https://doi.org/10.1007/s10231-021-01157-8
Publication status	Published - Jun 2022

Bibliographical note

Funding Information:
Open access funding provided by Università degli Studi di Udine within the CRUI-CARE Agreement. This paper was partially supported by PRID MARFAP, Dipartimento di Matematica Informatica e Fisica - Università di Udine, and PRIN 2017–2020 “Teoria dei gruppi e applicazioni.”

Keywords

Axial algebras
Baric algebras
Finite simple groups
Jordan algebras
Monster group

ASJC Scopus subject areas

Applied Mathematics

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FranchiC2021infiniteFinal published version, 1.6 MBLicence: Creative Commons: Attribution (CC BY)

Cite this

@article{e4ceadcca4e84c96a18b48ad41765aea,

title = "An infinite-dimensional 2-generated primitive axial algebra of Monster type",

abstract = "Rehren proved in Axial algebras. Ph.D. thesis, University of Birmingham (2015), Trans Am Math Soc 369:6953–6986 (2017) that a primitive 2-generated axial algebra of Monster type (α, β) , over a field of characteristic other than 2, has dimension at most 8 if α∉ \{ 2 β, 4 β\}. In this note, we show that Rehren{\textquoteright}s bound does not hold in the case α= 4 β by providing an example (essentially the unique one) of an infinite-dimensional 2-generated primitive axial algebra of Monster type (2,12) over an arbitrary field F of characteristic other than 2 and 3. We further determine its group of automorphisms and describe some of its relevant features.",

keywords = "Axial algebras, Baric algebras, Finite simple groups, Jordan algebras, Monster group",

author = "Clara Franchi and Mario Mainardis and Sergey Shpectorov",

note = "Funding Information: Open access funding provided by Universit{\`a} degli Studi di Udine within the CRUI-CARE Agreement. This paper was partially supported by PRID MARFAP, Dipartimento di Matematica Informatica e Fisica - Universit{\`a} di Udine, and PRIN 2017–2020 “Teoria dei gruppi e applicazioni.” ",

year = "2022",

month = jun,

doi = "10.1007/s10231-021-01157-8",

language = "English",

volume = "201",

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T1 - An infinite-dimensional 2-generated primitive axial algebra of Monster type

AU - Franchi, Clara

AU - Mainardis, Mario

AU - Shpectorov, Sergey

N1 - Funding Information: Open access funding provided by Università degli Studi di Udine within the CRUI-CARE Agreement. This paper was partially supported by PRID MARFAP, Dipartimento di Matematica Informatica e Fisica - Università di Udine, and PRIN 2017–2020 “Teoria dei gruppi e applicazioni.”

PY - 2022/6

Y1 - 2022/6

N2 - Rehren proved in Axial algebras. Ph.D. thesis, University of Birmingham (2015), Trans Am Math Soc 369:6953–6986 (2017) that a primitive 2-generated axial algebra of Monster type (α, β) , over a field of characteristic other than 2, has dimension at most 8 if α∉ { 2 β, 4 β}. In this note, we show that Rehren’s bound does not hold in the case α= 4 β by providing an example (essentially the unique one) of an infinite-dimensional 2-generated primitive axial algebra of Monster type (2,12) over an arbitrary field F of characteristic other than 2 and 3. We further determine its group of automorphisms and describe some of its relevant features.

AB - Rehren proved in Axial algebras. Ph.D. thesis, University of Birmingham (2015), Trans Am Math Soc 369:6953–6986 (2017) that a primitive 2-generated axial algebra of Monster type (α, β) , over a field of characteristic other than 2, has dimension at most 8 if α∉ { 2 β, 4 β}. In this note, we show that Rehren’s bound does not hold in the case α= 4 β by providing an example (essentially the unique one) of an infinite-dimensional 2-generated primitive axial algebra of Monster type (2,12) over an arbitrary field F of characteristic other than 2 and 3. We further determine its group of automorphisms and describe some of its relevant features.

KW - Axial algebras

KW - Baric algebras

KW - Finite simple groups

KW - Jordan algebras

KW - Monster group

UR - https://www.scopus.com/pages/publications/85115302650

U2 - 10.1007/s10231-021-01157-8

DO - 10.1007/s10231-021-01157-8

M3 - Article

AN - SCOPUS:85115302650

SN - 0373-3114

VL - 201

SP - 1279

EP - 1293

JO - Annali di Matematica Pura ed Applicata

JF - Annali di Matematica Pura ed Applicata

ER -

An infinite-dimensional 2-generated primitive axial algebra of Monster type

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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