Groebner-Shirshov bases for dialgebras

TL;DR

This paper introduces Groebner-Shirshov bases for dialgebras, providing foundational tools and normal forms for various algebraic structures including Leibniz algebras and Clifford dialgebras.

Contribution

It extends the theory of Groebner-Shirshov bases to dialgebras and establishes the Composition-Diamond lemma in this context.

Findings

Groebner-Shirshov bases constructed for multiple dialgebra types

Normal forms derived for complex algebraic structures

Unified approach to enveloping algebras and extensions

Abstract

In this paper, we define the Gr\"obner-Shirshov basis for a dialgebra. The Composition-Diamond lemma for dialgebras is given then. As results, we give Gr\"obner-Shirshov bases for the universal enveloping algebra of a Leibniz algebra, the bar extension of a dialgebra, the free product of two dialgebras, and Clifford dialgebra. We obtain some normal forms for algebras mentioned the above.