Novikov dialgebras and perm algebras

TL;DR

This paper explores the structure of Novikov dialgebras and perm algebras, introducing new operations, constructing bases, and identifying defining identities within algebraic varieties.

Contribution

It introduces a new operation on Perm algebras with derivation, constructs a basis for free Novikov dialgebras, and characterizes the identities of the resulting algebraic variety.

Findings

Constructed a linear basis for free Novikov dialgebras.

Proved that the class of algebras with the new operation forms a variety.

Identified the defining identities of this algebraic variety.

Abstract

In this paper, we consider Perm algebra with the derivation $d$. The algebra itself is equipped with the new operation $a\succ b = d(a) b$. We construct a linear basis of the free Novikov dialgebra in terms of new operations. Also, we prove that the class of algebras under the new operation form a variety. Finally, we find the defining identities of the variety.