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

作者：Tkachev Vladimir G^*

来源：Journal of Algebra, 2018, 504: 432-439.

DOI：10.1016/j.jalgebra..2018.02.031

• Commutative nonassociative algebras
• Griess algebra
• Idempotents
• Associative bilinear form
• Metrised algebras

In their paper of 1993, Meyer and Neutsch established the existence of a 48-dimensional associative subalgebra in the Griess algebra 0. By exhibiting an explicit counter example, the present paper shows a gap in the proof one of the key results in Meyer and Neutsch's paper, which states that an idempotent a in the Griess algebra is indecomposable if and only its Peirce 1-eigenspace (i.e. the 1-eigenspace of the linear transformation L. : x H ax) is one-dimensional. The present paper fixes this gap, and shows a more general result: let V be a real commutative nonassociative algebra with an associative inner product, and let c be a nonzero idempotent of V such that its Peirce 1-eigenspace is a subalgebra; then, c is indecomposable if and only if its Peirce 1-eigenspace is one-dimensional. The proof of this result is based on a general variational argument for real commutative metrised algebras with inner product.

• 出版日期2018-6-15