Split spin factor algebras

J. McInroy; S. Shpectorov

doi:10.1016/j.jalgebra.2021.12.022

Split spin factor algebras

J. McInroy^*, S. Shpectorov^*

^*Corresponding author for this work

Research output: Contribution to journal › Article (Academic Journal) › peer-review

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Abstract

Motivated by Yabe's classification of symmetric $2$-generated axial algebras of Monster type, we introduce a large class of algebras of Monster type $(\alpha, \frac{1}{2})$, generalising Yabe's $\mathrm{III}(\alpha,\frac{1}{2}, \delta)$ family. Our algebras bear a striking similarity with Jordan spin factor algebras with the difference being that we asymmetrically split the identity as a sum of two idempotents. We investigate the properties of this algebra, including the existence of a Frobenius form and ideals. In the $2$-generated case, where our algebra is isomorphic to one of Yabe's examples, we use our new viewpoint to identify the axet, that is, the closure of the two generating axes.

Original language	English
Pages (from-to)	380-397
Number of pages	18
Journal	Journal of Algebra
Volume	595
Early online date	22 Dec 2021
DOIs	https://doi.org/10.1016/j.jalgebra.2021.12.022
Publication status	Published - 1 Apr 2022

Bibliographical note

Funding Information:
The work of the second author has been supported by the Mathematical Center in Akademgorodok under the agreement No. 075-15-2019-1675 with the Ministry of Science and Higher Education of the Russian Federation .

Publisher Copyright:
© 2021 Elsevier Inc.

Keywords

Spin factor
Jordan algebra
Axial algebra
Monster type
2-generated
Non-associative
Idempotent

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10.1016/j.jalgebra.2021.12.022

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Accepted author manuscript, 285 KBLicence: CC BY-NC-ND

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@article{db53e40d3ca04155992acf92ace377d6,

title = "Split spin factor algebras",

abstract = "Motivated by Yabe's classification of symmetric \$2\$-generated axial algebras of Monster type, we introduce a large class of algebras of Monster type \$(\textbackslash{}alpha, \textbackslash{}frac\{1\}\{2\})\$, generalising Yabe's \$\textbackslash{}mathrm\{III\}(\textbackslash{}alpha,\textbackslash{}frac\{1\}\{2\}, \textbackslash{}delta)\$ family. Our algebras bear a striking similarity with Jordan spin factor algebras with the difference being that we asymmetrically split the identity as a sum of two idempotents. We investigate the properties of this algebra, including the existence of a Frobenius form and ideals. In the \$2\$-generated case, where our algebra is isomorphic to one of Yabe's examples, we use our new viewpoint to identify the axet, that is, the closure of the two generating axes. ",

keywords = "Spin factor, Jordan algebra, Axial algebra, Monster type, 2-generated, Non-associative, Idempotent",

author = "J. McInroy and S. Shpectorov",

note = "Funding Information: The work of the second author has been supported by the Mathematical Center in Akademgorodok under the agreement No. 075-15-2019-1675 with the Ministry of Science and Higher Education of the Russian Federation . Publisher Copyright: {\textcopyright} 2021 Elsevier Inc.",

year = "2022",

month = apr,

day = "1",

doi = "10.1016/j.jalgebra.2021.12.022",

language = "English",

volume = "595",

pages = "380--397",

journal = "Journal of Algebra",

issn = "0021-8693",

publisher = "Academic Press Inc.",

}

TY - JOUR

T1 - Split spin factor algebras

AU - McInroy, J.

AU - Shpectorov, S.

N1 - Funding Information: The work of the second author has been supported by the Mathematical Center in Akademgorodok under the agreement No. 075-15-2019-1675 with the Ministry of Science and Higher Education of the Russian Federation . Publisher Copyright: © 2021 Elsevier Inc.

PY - 2022/4/1

Y1 - 2022/4/1

N2 - Motivated by Yabe's classification of symmetric $2$-generated axial algebras of Monster type, we introduce a large class of algebras of Monster type $(\alpha, \frac{1}{2})$, generalising Yabe's $\mathrm{III}(\alpha,\frac{1}{2}, \delta)$ family. Our algebras bear a striking similarity with Jordan spin factor algebras with the difference being that we asymmetrically split the identity as a sum of two idempotents. We investigate the properties of this algebra, including the existence of a Frobenius form and ideals. In the $2$-generated case, where our algebra is isomorphic to one of Yabe's examples, we use our new viewpoint to identify the axet, that is, the closure of the two generating axes.

AB - Motivated by Yabe's classification of symmetric $2$-generated axial algebras of Monster type, we introduce a large class of algebras of Monster type $(\alpha, \frac{1}{2})$, generalising Yabe's $\mathrm{III}(\alpha,\frac{1}{2}, \delta)$ family. Our algebras bear a striking similarity with Jordan spin factor algebras with the difference being that we asymmetrically split the identity as a sum of two idempotents. We investigate the properties of this algebra, including the existence of a Frobenius form and ideals. In the $2$-generated case, where our algebra is isomorphic to one of Yabe's examples, we use our new viewpoint to identify the axet, that is, the closure of the two generating axes.

KW - Spin factor

KW - Jordan algebra

KW - Axial algebra

KW - Monster type

KW - 2-generated

KW - Non-associative

KW - Idempotent

U2 - 10.1016/j.jalgebra.2021.12.022

DO - 10.1016/j.jalgebra.2021.12.022

M3 - Article (Academic Journal)

SN - 0021-8693

VL - 595

SP - 380

EP - 397

JO - Journal of Algebra

JF - Journal of Algebra

ER -

Split spin factor algebras

Abstract

Bibliographical note

Keywords

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