On the classification of 2-generated axial algebras of Majorana type

Yabe, Takahiro

doi:10.48550/arxiv.2008.01871

Cited by 8 publications

(42 citation statements)

References 7 publications

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“…Combining our reduction result and Yabe's classification of 2-generated symmetric M(α, β)-axial algebras [18], we are left only to check which of the 2-generated algebras have a finite axet and which have property (J). It is known in folklore that algebras of Jordan type β = 1 2 generically have an infinite axet, but it can be finite for some parameters.…”

Section: Introductionmentioning

confidence: 94%

“…Apart from the two exceptional cases, A is either a direct sum of a (symmetric) 2-generated algebra with a 1A algebra, or a symmetric 2-generated algebra on a X(2n) axet. While this paper was in preparation, the symmetric 2generated algebras were classified by Yabe [18]. Notice that different cases of this theorem impose different restrictions on the algebra.…”

Section: Identifying Shapes On 3-generated Axetsmentioning

confidence: 99%

“…A symmetric 2-generated axial algebra A = a, b is one where there is an involutory automorphism f , called the flip, which switches a and b. For the generalised Monster fusion law M(α, β), such algebras were first considered by Rehren in [16] and then classified by Yabe in [18] with one of the cases finished by Franchi and Mainardis in [3]. A typical symmetric M(α, β)axial algebra A does not exists for all values of α and β, but rather for some variety in α and β (and sometimes depends on an extra parameter).…”

Section: Symmetric 2-generated M(α β)-Axial Algebrasmentioning

confidence: 99%

“…1. an axial algebra of Jordan type α, or β; 2. a quotient of the highwater algebra H, or its characteristic 5 cover Ĥ, where (α, β) = (2, 1 2 ); or 3. one of the algebras listed in [18,Table 2].…”

Section: Symmetric 2-generated M(α β)-Axial Algebrasmentioning

confidence: 99%

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From forbidden configurations to a classification of some axial algebras of Monster type

McInroy¹,

Shpectorov²

2021

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Axial algebras are a recently introduced class of non-associative algebras, where the focus is on special idempotents called axes and the properties of the algebra are specified in terms of a fusion law which governs how eigenvectors of an axis multiply. Jordan algebras and Majorana algebras (including the Griess algebra for the Monster) are examples.Ivanov introduced the shape of a Majorana algebra, which is the configuration of 2-generated subalgebras arising in that algebra. In order to generalise this concept to arbitrary axial algebras and to free it from the necessity of the ambient algebra, we introduce the concept of an axet and shapes on an axet. A shape can be viewed as an algebra version of a group amalgam. Just like a group amalgam has a universal completion, a shape leads to a unique algebra completion which may be non-trivial or it may collapse. In fact, in an earlier project, we observed that the great majority of shapes collapse. We are looking for small forbidden configurations, i.e. small collapsing shapes, as tools to establish non-existence of larger algebras. In this paper, we investigate whole families of shapes for the generalised Monster type fusion law M(α, β) and we show that almost all of them collapse. We also identify all the exceptions and classify their completions.

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Section: Introductionmentioning

confidence: 94%

Section: Identifying Shapes On 3-generated Axetsmentioning

confidence: 99%

Section: Symmetric 2-generated M(α β)-Axial Algebrasmentioning

confidence: 99%

Section: Symmetric 2-generated M(α β)-Axial Algebrasmentioning

confidence: 99%

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From forbidden configurations to a classification of some axial algebras of Monster type

McInroy¹,

Shpectorov²

2021

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“…Recently Yabe [10] classified symmetric 2-generated axial algebras of Monster type. In doing so, he introduced several new families of 2-generated algebras, in addition to those found by Rehren in [9] and found by Joshi using the double axis construction [6] (see also [3]).…”

Section: Introductionmentioning

confidence: 99%

Split spin factor algebras

McInroy¹,

Shpectorov

2021

Preprint

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Motivated by Yabe's classification of symmetric 2-generated axial algebras of Monster type [10], we introduce a large class of algebras of Monster type (α, 1 2 ), generalising Yabe's III(α, 1 2 , δ) 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.

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2-generated axial algebras of Monster type

Franchi¹,

Mainardis²,

Shpectorov³

2021

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In this paper we provide the basic setup for a project, initiated by Felix Rehren in [16], aiming at classifying all 2-generated primitive axial agebras of Monster type (α, β). We first revise Rehren's construction of an initial object in the cathegory of primitive n-generated axial algebras giving a formal one, filling some gaps and, though correcting some inaccuracies, confirm Rehren's results. Then we focus on 2-generated algebras which naturally part into three cases: the two critical cases α = 2β and α = 4β, and the generic case (i.e. all the rest). About these cases, which will be dealt in detail in subsequent papers, we give bounds on the dimensions (the generic case already treated by Rehen) and classify all 2-generated primitive axial algebras of Monster type (α, β) over Q(α, β) for α and β algebraically independent indeterminates over Q. Finally we restrict to the 2-generated Majorana algebras (i.e. when α = 1 4 and β = 1 32 ), showing that these fall precisely into the nine isomorphism types of the Norton-Sakuma algebras.

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On the classification of 2-generated axial algebras of Majorana type

Cited by 8 publications

References 7 publications

From forbidden configurations to a classification of some axial algebras of Monster type

From forbidden configurations to a classification of some axial algebras of Monster type

Split spin factor algebras

2-generated axial algebras of Monster type

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