On a variety of right-symmetric algebras

TL;DR

The paper investigates the Specht property for the variety of right-symmetric algebras by constructing a finite-dimensional metabelian right-symmetric algebra that does not admit a finite basis of identities. It defines the variety $\mathcal{R}$ with specific identities, builds algebras $ P_n $ within this variety, and analyzes the structure of free algebras via an operator-theoretic framework, introducing $V_{x,y}=L_xR_y$ and studying the relationships between polynomial identities and operator identities. The main result shows that $P_2$ has no finite basis of identities, demonstrating that the Specht property fails for this class of algebras. The methods combine L'vov–Isaev style constructions, linearization techniques, and a matrix-operator viewpoint ($E_0(P_n)\cong M_n(\mathbb{F})$) to establish the non-Specht nature of the right-symmetric variety in question, highlighting limitations of finite basis properties beyond associative and Lie contexts.

Abstract

We construct a finite-dimensional metabelian right-symmetric algebra over an arbitrary field that does not have a finite basis of identities.

Theorems & Definitions (34)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof