A note on 2-generated symmetric axial algebras of Monster type

Abstract

In [10], Yabe gives an almost complete classification of primitive symmetric 2-generated axial algebras of Monster type. In this note, we construct a new infinite-dimensional primitive 2-generated symmetric axial algebra of Monster type (2, 12 ) over a field of characteristic 5, and use this algebra to complete the last case left open in Yabe's classification.

On primitive axial decomposition algebras of Majorana type with degenerate eigenvalues

“…When ξ = η, the condition dihedral is equivalent to being 2-generated and having an automorphism called flip, which switches a 0 and a 1 . So the classification for ξ = η reduces to the results of our previous work [7] if ch F = 5 and to [2] and [7] if ch F = 5. The main result of this paper is the classification of dihedral axial decomposition algebras of Majorana type (η, η) with η = 1 2 .…”

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

“…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).…”

“…There are two families of finite-dimensional algebra on Yabe's list that generically have the axet X(∞), namely IY 3 (α, 1 2 , µ) 3 and IY 5 (α, 1 2 ). Since β = 1 2 for both these algebras, we assume that char(F) = 2.…”

“…It is an infinite dimensional 2-generated symmetric M(2, 1 2 )-axial algebra with axet X(∞) over any field of characteristic not 2, or 3. In characteristic 5, the highwater algebra has a cover Ĥ which was introduced by Franchi and Mainardis [3]. We give the multiplication for H and Ĥ in Table 2.…”

“…Table 2: The highwater algebra and its cover From [5], in H, the Miyamoto involution τ k := τ a k , for k ∈ Z, acts by fixing all the σ j and mapping a i → a 2k−i . For Ĥ, τ k acts on the a i in the same way and fixes s 0,j if j ∈ 3N, but when 3|j it maps s ā,j → s 2k−a,j [3].…”