Associative subalgebras of low-dimensional Majorana algebras

Abstract

A Majorana algebra is a commutative nonassociative real algebra generated by a finite set of idempotents, called Majorana axes, that satisfy some of the properties of the 2A-axes of the Monster Griess algebra. The term was introduced by A.A. Ivanov in 2009 inspired by the work of S. Sakuma and M. Miyamoto. In the present paper, we revisit Mayer and Neutsch's theorem on associative subalgebras of the Griess algebra in the context of Majorana theory. We apply this result to determine all the maximal associative subalgebras of some low-dimensional Majorana algebras; namely, the Majorana algebras generated by two Majorana axes and the Majorana representations of the symmetric group of degree 4 involving 3C-algebras.

A correction of the decomposability result in a paper by Meyer–Neutsch

“…One of the key results claimed in [5], Theorem 11 asserts that an idempotent c ∈ G is indecomposable if and only if the Peirce eigenspace G c (1) with the eigenvalue 1 is at most one-dimensional. This result was used by Miyamoto (see Theorem 6.8 in [6]) and in a later research on associative subalgebras of low-dimensional Majorana algebras, see for instance p. 576 in [2] and section 3.1 in [4].…”

Code algebras, axial algebras and VOAs

“…However, this includes all basis elements, so x = 0 and the standard torus is maximal. In the context of Majorana algebras, maximal tori have been studied and classified for low-dimensional cases [2].…”

On an extremal property of Jordan algebras of Clifford type

“…A more recent example which also appears in this context is the class of nonassociative algebras decoding the geometric structure of cubic minimal cones and cubic polynomial solutions to certain elliptic PDEs [13,Chapter 6], [12], [19], [20]. Some related questions as well as geometry of idempotents are also discussed in [17], [5], [6], [2]. It is known [20], [10] (see also Proposition 2.1 below) that if V is a Euclidean metrised algebra then the set of nonzero idempotents Idm(V ) is nonempty and there exists an idempotent c = 0 such that (2) |c| 2 ≤ |c ′ | 2 , ∀c ′ ∈ Idm(V ),…”

“…Some related questions as well as geometry of idempotents are also discussed in [17], [5], [6], [2]. It is known [20], [10] (see also Proposition 2.1 below) that if V is a Euclidean metrised algebra then the set of nonzero idempotents Idm(V ) is nonempty and there exists an idempotent c = 0 such that (2) |c| 2 ≤ |c ′ | 2 , ∀c ′ ∈ Idm(V ),…”