Axes in non-associative algebras

Louis Rowen, Yoav Segev

Research output: Contribution to journalArticlepeer-review

Abstract

“Fusion rules” are laws of multiplication among eigenspaces of an idempotent. This terminology is relatively new and is closely related to axial algebras, introduced recently by Hall, Rehren and Shpectorov. Axial algebras, in turn, are closely related to 3 -transposition groups and Vertex operator algebras. In this paper we consider fusion rules for semisimple idempotents, following Albert in the power-associative case. We examine the notion of an axis in the non-commutative setting and show that the dimension d of any algebra A generated by a pair a, b of (not necessarily Jordan) axes of respective types (λ, δ) and (λ, δ) must be at most 5; d cannot be 4. If d ≤ 3 we list all the possibilities for A up to isomorphism. We prove a variety of additional results and mention some research questions at the end.

Original languageEnglish
Pages (from-to)2366-2381
Number of pages16
JournalTurkish Journal of Mathematics
Volume45
Issue number6
DOIs
StatePublished - 1 Jan 2021

Keywords

  • Axial algebra
  • Axis
  • Flexible algebra
  • Fusion rule
  • Idempotent
  • Jordan type
  • Power-associative

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