On primitive 3-generated axial algebras of Jordan type

Gorshkov, Ilya; Staroletov, Alexey

doi:10.1016/j.jalgebra.2020.07.014

Cited by 9 publications

(6 citation statements)

References 17 publications

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“…The 3-generated case has also been classified by Gorshkov and Staroletov [27], namely every axial algebra of Jordan type 1 2 is covered by a 9-dimensional Jordan algebra from a 4-parameter family. Generically, they are isomorphic to the 3 × 3 matrix algebra.…”

Section: Theorem 44 ([33]mentioning

confidence: 99%

Axial algebras of Jordan and Monster type

McInroy¹,

Shpectorov²

2022

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Axial algebras are a class of non-associative commutative algebras whose properties are defined in terms of a fusion law. When this fusion law is graded, the algebra has a naturally associated group of automorphisms and thus axial algebras are inherently related to group theory. Examples include most Jordan algebras and the Griess algebra for the Monster sporadic simple group.In this survey, we introduce axial algebras, discuss their structural properties and then concentrate on two specific classes: algebras of Jordan and Monster type, which are rich in examples related to simple groups.

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Section: Theorem 44 ([33]mentioning

confidence: 99%

Axial algebras of Jordan and Monster type

McInroy¹,

Shpectorov²

2022

Preprint

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“…Then, every Jordan algebra generated by its idempotent elements is J ( ); the answer has been proved affirmative for algebras generated by two or three primitive axes [16,19]. Studying central extensions of Jordan algebras could be a way to delve into this question.…”

Section: Axial Central Extensions Of Jordan Algebrasmentioning

confidence: 99%

“…It is obvious that there are not non-split J ( 12 )-axial central extensions of the axial Jordan algebra of dimension 1. Also, by [16,19], we know that the theorem holds true for every axial Jordan algebra generated by two or three primitive axes; then we use the help of Lemma 3.9 to identify them. Indeed, in the next Table 50 we display such algebras J together with a set of primitive generating axes X.…”

Section: Axial Central Extensions Of Jordan Algebrasmentioning

confidence: 99%

“…Also, a classification of primitive symmetric two-generated axial algebras of Monster type was given between Yabe's [40], Franchi and Mainardis' [13] and Franchi, Mainardis and McInroy's [10]. On the other hand, Gorshkov and Staroletov showed that a three-generated primitive axial algebra of Jordan type has dimension at most 9 [16]; Khasraw, McInroy and Shpectorov enumerated all the three-generated primitive axial algebras of Monster type (α, β) of a certain subclass, the so-called 4-algebras [26].…”

Section: Introductionmentioning

confidence: 99%

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Central extensions of axial algebras

Kaygorodov¹,

González²,

Páez-Guillán³

2022

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In this article, we develop a further adaptation of the method of Skjelbred-Sund to construct central extensions of axial algebras. We use our method to prove that all axial central extensions (with respect to a maximal set of axes) of complex simple finite-dimensional Jordan algebras are split, and that all non-split axial central extensions of dimension n ≤ 4 over an algebraically closed field of characteristic not 2 are Jordan. Also, we give a classification of 2-dimensional axial algebras and describe some important properties of these algebras.

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“…For 3-generated algebras, this question was answered affirmatively in the recent paper [GS20]: such algebras are at most 9-dimensional.…”

Section: Introductionmentioning

confidence: 96%