Polynomial identities for tangent algebras of monoassociative loops

TL;DR

This paper introduces degree n Sabinin algebras, characterizing tangent structures of monoassociative loops through polynomial identities, and explores their properties and relations to known algebraic systems.

Contribution

It defines degree n Sabinin algebras, characterizes degree 4 cases, and identifies a new degree 5 identity, advancing the understanding of tangent algebras of monoassociative loops.

Findings

Degree 4 Sabinin algebras characterized by specific polynomial identities.

One of the quaternators in degree 4 is redundant.

A new degree 5 identity distinct from lower degrees.

Abstract

We introduce degree n Sabinin algebras, which are defined by the polynomial identities up to degree n in a Sabinin algebra. Degree 4 Sabinin algebras can be characterized by the polynomial identities satisfied by the commutator, associator and two quaternators in the free nonassociative algebra. We consider these operations in a free power associative algebra and show that one of the quaternators is redundant. The resulting algebras provide the natural structure on the tangent space at the identity element of an analytic loop for which all local loops satisfy monoassociativity, a^2 a = a a^2. These algebras are the next step beyond Lie, Malcev, and Bol algebras. We also present an identity of degree 5 which is satisfied by these three operations but which is not implied by the identities of lower degree.