Split spin factor algebras

Abstract

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.

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

“…Note that the Miyamoto involutions are minus reflections in the underlying 2-dimensional quadratic space. As we will see later, this more general setup of a 2-dimensional reflection group arises also in the split spin factor algebras [15] and we briefly describe it here.…”

“…Definition 7.4. [15] Let E be a vector space with a symmetric bilinear form b and α ∈ F. The split spin factor algebra S(b, α) is the algebra on E ⊕ Fz 1 ⊕ Fz 2 with multiplication…”

“…From [15], we have the following properties. If b(e, e) = 1, then x := 1 2 (e + αz 1 + (α + 1)z 2 ) is a (primitive) axis of Monster type M(α, 1 2 ) and these are all such axes (not counting z 1 , which is of Jordan type α).…”

“…Indeed for α = −1 and µ = 1, S(b, −1) • := x, y has codimension 1. In [15], we introduce a nil cover of this subalgebra. Definition 7.6.…”

“…We may now consider the axets for both of these algebras simultaneously. From [15], the Miyamoto involution for x is τ x = −r e (cf. Section 5.2).…”