Central orders in simple right-alternative superalgebras and right-symmetric algebras

TL;DR

This work analyzes central orders in two nonassociative families—simple finite-dimensional right-symmetric algebras and simple right-alternative superalgebras—demonstrating that each central order embeds into a finitely generated module over its center (or the even part of the center in the supercase). By examining explicit constructions of RS-algebras (including matrix-type and $V_2+M_2(F)$-type algebras) and abelian-type/asymmetric double right-alternative superalgebras (e.g., $B_{n|n}$, $B_{2|2}(\nu)$, $B_{4|4}(w)$), the paper proves finiteness properties for their central orders via explicit localization arguments (e.g., showing $Bz^k$ resides in a finite $Z$-span of basis elements). The results extend classical central-order theory to new nonassociative contexts and yield structural decompositions of localized algebras $Z^{-1}B$, with corollaries describing when $Z^{-1}B$ is associative or falls into known Shestakov-type algebras. Overall, the work tightens the understanding of finiteness and embedding properties for central orders in RS and right-alternative superalgebras, connecting them to established algebraic frameworks and classifications.

Abstract

We consider some recently constructed examples of simple finite-dimensional right-alternative superalgebras and right-symmetric algebras. We prove that the central order in any of these algebras and superalgebras is embedded in a finite module over its center (or over the even part of its center in the case of superalgebras)

Theorems & Definitions (14)

• Lemma 1
• proof
• Lemma 2
• proof
• Theorem 1
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5