Algebraic deformation quantization of Leibniz algebras

TL;DR

This paper develops a deformation quantization framework for Leibniz algebras using rack bialgebras, providing explicit formulas and establishing a deformation theory that generalizes previous constructions.

Contribution

It introduces canonical rack bialgebras for Leibniz algebras and formulates a general deformation theory, leading to explicit rack-star-product formulas.

Findings

Constructed canonical rack bialgebras for any Leibniz algebra.

Derived explicit formulas for the rack-star-product.

Established a deformation theory for rack bialgebras.

Abstract

In this paper we focus on a certain self-distributive multiplication on coalgebras, which leads to so-called rack bialgebra. We construct canon-ical rack bialgebras (some kind of enveloping algebras) for any Leibniz algebra. Our motivation is deformation quantization of Leibniz algebras in the sense of [6]. Namely, the canonical rack bialgebras we have constructed for any Leibniz algebra lead to a simple explicit formula of the rack-star-product on the dual of a Leibniz algebra recently constructed by Dherin and Wagemann in [6]. We clarify this framework setting up a general deformation theory for rack bialgebras and show that the rack-star-product turns out to be a deformation of the trivial rack bialgebra product.