On the universal Drinfeld-Yetter algebra

TL;DR

This paper provides an explicit formula for the structure constants of the universal Drinfeld-Yetter algebra using diagrammatic representations called Drinfeld-Yetter looms, advancing understanding of quantization of Lie bialgebras.

Contribution

It introduces a concrete diagrammatic formula for the algebra's structure constants, clarifying its structure and role in Lie bialgebra quantization.

Findings

Explicit formula for structure constants using Drinfeld-Yetter looms

Enhanced understanding of the algebra's role in quantization

Diagrammatic approach simplifies complex algebraic relations

Abstract

The universal Drinfeld-Yetter algebra is an associative algebra whose co-Hochschild cohomology controls the existence of quantization functors of Lie bialgebras, such as the renowned one due to Etingof and Kazhdan. It was initially introduced by Enriquez and later re-interpreted by Appel and Toledano Laredo as an algebra of endomorphisms in the colored PROP of a Drinfeld-Yetter module over a Lie bialgebra. In this paper, we provide an explicit formula for its structure constants in terms of certain diagrams, which we term Drinfeld-Yetter looms.