Double axes and subalgebras of Monster type in Matsuo algebras

Galt, Alexey; Joshi, Vijay; Mamontov, A. S.; Shpectorov, Sergey; Staroletov, Alexey

doi:10.1080/00927872.2021.1917589

Cited by 16 publications

(15 citation statements)

References 23 publications

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“…α(5α−2) q. For 6J(α, α 2 ), we have u = âi + âi+3 − 2 α âiâi+3, w = 2(âi + âi+1) − 4 α âiâi+1 as per the basis in [6]. For 6Y( 1 2 , 2), we have a4 = a0 + a2 − a0a2, d = a−1 − a2, z = q.…”

Section: Algebras With Axet X(6)mentioning

confidence: 99%

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

McInroy¹,

Shpectorov²

2021

Preprint

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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: Algebras With Axet X(6)mentioning

confidence: 99%

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

McInroy¹,

Shpectorov²

2021

Preprint

Self Cite

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show abstract

“…(3) the algebras V 5 (β) and V 8 (β) defined in [5]. (4) the 3-dimensional algebra V 3 (β) with basis (ā 0 , ā1 , ā2 ) and the multiplication defined as in Table 2.…”

Section: The Symmetric Casementioning

confidence: 99%

“…(2) Proof. By the remark after Theorem 4.1, V is determined, up to homomorphic images, by the pair (λ 1 , λ 2 ), which must be a solution of (5). By [3,Corollary 3.8 ] and Proposition 3.7, V is spanned on F by the set ā−2 , ā−1 , ā0 , ā1 ,ā 2 ,ā 3 , s0,1 , and s0,3 .…”

Section: In Which Case It Has a Unique Non Trivial Quotient Over The ...mentioning

confidence: 99%

“…This class was introduced by J. Hall, F. Rehren and S. Shpectorov [8] in order to axiomatise some key features of many important classes of algebras, such as the weight-2 components of OZ-type vertex operator algebras, Jordan algebras and Matsuo algebras (see the introductions of [8], [15] and [5]). They are also of particular interest for finite group theorists as most of the finite simple groups, or their automorphs, can be faithfully and effectively represented as automorphism groups of these algebras.…”

Section: Introductionmentioning

confidence: 99%

“…In Section 6, we classify the non-symmetric algebras. In [5], A. Galt, V. Joshi, A. Mamontov, S. Shpectorov and A. Staroletov introduced the concept of double axis in a Matsuo algebra M η (Γ). They proved that double axes satisfy the fusion law M(2η, η) and classified the primitive subalgebras of M η (Γ) generated by two double axes or by an axis and a double axis: besides the algebras of Jordan type 1A, 2B, 3C(η) and 3C(2η), they found three new algebras of dimensions 4, 5 and 8 which we refer to by Q 2 (η), V 5 (η), and V 8 (η), respectively.…”

Section: Introductionmentioning

confidence: 99%

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