Modules over Axial Algebras

Medts, Tom De; Couwenberghe, Michiel Van

doi:10.1007/s10468-018-9844-y

Cited by 5 publications

(1 citation statement)

References 10 publications

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“…In this paper, we explore when code algebras are also axial algebras and classify when these have a particularly symmetric multiplicative structure, namely that the fusion law is Z 2 -graded. Axial algebras are a new class of commutative non-associative algebras that has attracted considerable interest recently (see [3,4,6,8,9,10,11,12]) since its introduction by Hall, Rehren and Shpectorov in [2]. The class includes several interesting algebras, in particular, subalgebras of the Greiss algebra, Majorana algebras, Jordan algebras and Matsuo algebras.…”

Section: Introductionmentioning

confidence: 99%

Code algebras which are axial algebras and their ℤ2-gradings

Castillo-Ramirez

McInroy

2019

Isr. J. Math.

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A code algebra AC is a non-associative commutative algebra defined via a binary linear code C. We study certain idempotents in code algebras, which we call small idempotents, that are determined by a single non-zero codeword. For a general code C, we show that small idempotents are primitive and semisimple and we calculate their fusion law. If C is a projective code generated by a conjugacy class of codewords, we show that AC is generated by small idempotents and so is, in fact, an axial algebra. Furthermore, we classify when the fusion law is Z2-graded. In doing so, we exhibit an infinite family of Z2 × Z2-graded axial algebras -these are the first known examples of axial algebras with a non-trivial grading other than a Z2-grading.

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Section: Introductionmentioning

confidence: 99%

Code algebras which are axial algebras and their ℤ2-gradings

Castillo-Ramirez

McInroy

2019

Isr. J. Math.

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Decomposition algebras and axial algebras

et al. 2020

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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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Matsuo algebras in characteristic 2

De Medts,

Stout

2024

Journal of Algebra

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Modules over Axial Algebras

Cited by 5 publications

References 10 publications

Code algebras which are axial algebras and their ℤ2-gradings

Code algebras which are axial algebras and their ℤ2-gradings

Decomposition algebras and axial algebras

Matsuo algebras in characteristic 2

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