On special identities for dialgebras

TL;DR

This paper explores special identities in dialgebras using operad theory, showing that all multilinear identities derive from those of related algebras, confirming a recent conjecture.

Contribution

It introduces a method to derive all multilinear special identities for dialgebras from identities of associated algebras, confirming a conjecture by Bremner, Felipe, and Sanchez-Ortega.

Findings

All polylinear special identities for dialgebras can be obtained from algebra identities.

The approach confirms a conjecture by Bremner, Felipe, and Sanchez-Ortega.

Operad-based method simplifies the study of identities in dialgebras.

Abstract

For every variety of algebras over a field, there is a natural definition of a corresponding variety of dialgebras (Loday-type algebras). In particular, Lie dialgebras are equivalent to Leibniz algebras. We use an approach based on the notion of an operad to study the problem of finding special identities for dialgebras. It is proved that all polylinear special identities for dialgebras can be obtained from special identities for corresponding algebras by means of a simple procedure. A particular case of this result confirms the conjecture by M. Bremner, R. Felipe, and J. Sanchez-Ortega, arXiv:1108.0586.