Decomposition algebras and axial algebras

Tom De Medts; Simon F. Peacock; Sergey Shpectorov; Michiel Van Couwenberghe

doi:10.1016/j.jalgebra.2020.02.027

Decomposition algebras and axial algebras

Tom De Medts, Simon F. Peacock, Sergey Shpectorov, Michiel Van Couwenberghe

Mathematics

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

2 Citations (Scopus)

190 Downloads (Pure)

Abstract

We introduce decomposition algebras as a natural generalization of axial algebras, Majorana algebras and the Griess algebra. They remedy three limitations of axial algebras:

(1) They separate fusion laws from specific values in a field, thereby allowing repetition of eigenvalues; (2) They allow for decompositions that do not arise from multiplication by idempotents; (3) They admit a natural notion of homomorphisms, making them into a nice category.

We exploit these facts to strengthen the connection between axial algebras and groups. In particular, we provide a definition of a universal Miyamoto group which makes this connection functorial under some mild assumptions.

We illustrate our theory by explaining how representation theory and association schemes can help to build a decomposition algebra for a given (permutation) group. This construction leads to a large number of examples.

We also take the opportunity to fix some terminology in this rapidly expanding subject.

Original language	English
Pages (from-to)	287-314
Number of pages	28
Journal	Journal of Algebra
Volume	556
Early online date	31 Mar 2020
DOIs	https://doi.org/10.1016/j.jalgebra.2020.02.027
Publication status	Published - 15 Aug 2020

Keywords

17A99, 20F29
Association schemes
Axial algebras
Decomposition algebras
Fusion laws
Griess algebra
Majorana algebras
Norton algebras
Representation theory
math.GR
math.RA

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10.1016/j.jalgebra.2020.02.027Licence: None: All rights reserved

De_Medts_et_al_Decomposition_algebras_and_axial_algebras_Journal_of_Algebra_2020Accepted author manuscript, 3.15 MBLicence: Creative Commons: Attribution-NonCommercial-NoDerivs (CC BY-NC-ND)

https://www.sciencedirect.com/science/article/pii/S0021869320301228Licence: None: All rights reserved

Cite this

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title = "Decomposition algebras and axial algebras",

abstract = "We introduce decomposition algebras as a natural generalization of axial algebras, Majorana algebras and the Griess algebra. They remedy three limitations of axial algebras: (1) They separate fusion laws from specific values in a field, thereby allowing repetition of eigenvalues; (2) They allow for decompositions that do not arise from multiplication by idempotents; (3) They admit a natural notion of homomorphisms, making them into a nice category. We exploit these facts to strengthen the connection between axial algebras and groups. In particular, we provide a definition of a universal Miyamoto group which makes this connection functorial under some mild assumptions. We illustrate our theory by explaining how representation theory and association schemes can help to build a decomposition algebra for a given (permutation) group. This construction leads to a large number of examples. We also take the opportunity to fix some terminology in this rapidly expanding subject. ",

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TY - JOUR

T1 - Decomposition algebras and axial algebras

AU - Medts, Tom De

AU - Peacock, Simon F.

AU - Shpectorov, Sergey

AU - Couwenberghe, Michiel Van

PY - 2020/8/15

Y1 - 2020/8/15

N2 - We introduce decomposition algebras as a natural generalization of axial algebras, Majorana algebras and the Griess algebra. They remedy three limitations of axial algebras: (1) They separate fusion laws from specific values in a field, thereby allowing repetition of eigenvalues; (2) They allow for decompositions that do not arise from multiplication by idempotents; (3) They admit a natural notion of homomorphisms, making them into a nice category. We exploit these facts to strengthen the connection between axial algebras and groups. In particular, we provide a definition of a universal Miyamoto group which makes this connection functorial under some mild assumptions. We illustrate our theory by explaining how representation theory and association schemes can help to build a decomposition algebra for a given (permutation) group. This construction leads to a large number of examples. We also take the opportunity to fix some terminology in this rapidly expanding subject.

AB - We introduce decomposition algebras as a natural generalization of axial algebras, Majorana algebras and the Griess algebra. They remedy three limitations of axial algebras: (1) They separate fusion laws from specific values in a field, thereby allowing repetition of eigenvalues; (2) They allow for decompositions that do not arise from multiplication by idempotents; (3) They admit a natural notion of homomorphisms, making them into a nice category. We exploit these facts to strengthen the connection between axial algebras and groups. In particular, we provide a definition of a universal Miyamoto group which makes this connection functorial under some mild assumptions. We illustrate our theory by explaining how representation theory and association schemes can help to build a decomposition algebra for a given (permutation) group. This construction leads to a large number of examples. We also take the opportunity to fix some terminology in this rapidly expanding subject.

KW - 17A99, 20F29

KW - Association schemes

KW - Axial algebras

KW - Decomposition algebras

KW - Fusion laws

KW - Griess algebra

KW - Majorana algebras

KW - Norton algebras

KW - Representation theory

KW - math.GR

KW - math.RA

UR - http://www.scopus.com/inward/record.url?scp=85082747862&partnerID=8YFLogxK

U2 - 10.1016/j.jalgebra.2020.02.027

DO - 10.1016/j.jalgebra.2020.02.027

M3 - Article

SN - 0021-8693

VL - 556

SP - 287

EP - 314

JO - Journal of Algebra

JF - Journal of Algebra

ER -

Decomposition algebras and axial algebras

Abstract

Keywords

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