Associative Subalgebras of the Griess Algebra

The group of symmetries of the shorter Moonshine module

“…There are two cases: Table 1 (cf. [7,14]). The components of X + have dimensions 1, 1, 4371 and 96255.…”

Associative subalgebras of low-dimensional Majorana algebras

“…The following results were obtained in [15] with V = V M ; however, their proof applies, without change, to Majorana algebras that satisfy M2 . Notice that if B := {x i ∈ V : 1 ≤ i ≤ k} is a set of pairwise orthogonal idempotents, then the associative subalgebra U := B coincides with the linear span of B; in particular, the dimension of U is k. If the idempotents of B are indecomposable, all the non-zero idempotents of U are finite sums of the idempotents of B; in particular, U may have at most one orthogonal basis of indecomposable idempotents.…”

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

“…In their 1993 paper [5], Meyer and Neutsch established the existence of a 48dimensional associative subalgebra in the Griess algebra G and conjectured that 48 is the largest possible dimension of an associative subalgebra of G. This conjecture has been proved by Miyamoto [6].…”

“…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].…”

“…Recall the statement of the theorem. Theorem 1.1 (Theorem 11, in [5]). An idempotent a ∈ G is indecomposable if and only if the Peirce eigenspace G a (1) = {x ∈ G : L a x = x} is at most onedimensional.…”