Universal Axial Algebras and a Theorem of Sakuma

Abstract

In the first half of this paper, we define axial algebras: nonassociative commutative algebras generated by axes, that is, semisimple idempotents---the prototypical example of which is Griess' algebra [C85] for the Monster group. When multiplication of eigenspaces of axes is controlled by fusion rules, the structure of the axial algebra is determined to a large degree. We give a construction of the universal Frobenius axial algebra on $n$ generators with a specified fusion rules, of which all $n$-generated Frobenius axial algebras with the same fusion rules are quotients. In the second half, we realise this construction in the Majorana / Ising / $\mathrm{Vir}(4,3)$-case on $2$ generators, and deduce a result generalising Sakuma's theorem in VOAs [S07].

24 pages; comments welcome

References in corpus (1)

• Primitive axial algebras of Jordan type

Cited by in corpus (21)

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• Decomposition algebras and axial algebras
• An expansion algorithm for constructing axial algebras
• The universality of one half in commutative nonassociative algebras with identities
• A note on 2-generated symmetric axial algebras of Monster type
• Modules over axial algebras
• Code algebras, axial algebras and VOAs
• The commutative nonassociative algebra of metric curvature tensors
• On the classification of 2-generated axial algebras of Majorana type
• Primitive 4-generated axial algebras of Jordan type
• Code algebras which are axial algebras and their ${\mathbb{Z}}_2$-gradings
• Inner isotopes associated with automorphisms of commutative associative algebras
• Miyamoto groups of code algebras
• V.M. Miklyukov: from dimension 8 to nonassociative algebras
• Killing metrized commutative nonassociative algebras associated with Steiner triple systems
• $2A$-Majorana Representations of $A_{12}$
• Split spin factor algebras
• Associative and Jordan algebras generated by two idempotents
• $3$-generated axial algebras with a minimal Miyamoto group
• Miyamoto involutions in axial algebras of Jordan type half
• On primitive axial decomposition algebras of Majorana type with degenerate eigenvalues