A First Sight Towards Primitively Generated Connected Braided Bialgebras

TL;DR

This paper explores the structure of primitively generated connected braided bialgebras, showing they can be viewed as universal enveloping algebras of their primitive elements, with applications to quadratic coalgebras.

Contribution

It establishes a framework linking braided bialgebras to universal enveloping algebras via Nichols algebras, providing new structural insights.

Findings

Connected braided bialgebras can be reconstructed as universal enveloping algebras of primitive elements.

The construction applies to bialgebras with quadratic associated graded coalgebras.

Provides a description of such bialgebras in terms of Nichols algebras.

Abstract

The main aim of this paper is to investigate the structure of primitively generated connected braided bialgebras $A$ with respect to the braided vector space $P$ consisting of their primitive elements. When the Nichols algebra of $P$ is obtained dividing out the tensor algebra $T(P) $ by the two-sided ideal generated by its primitive elements of degree at least two, we show that $A$ can be recovered as a sort of universal enveloping algebra of $P$. One of the main applications of our construction is the description, in terms of universal enveloping algebras, of connected braided bialgebras whose associated graded coalgebra is a quadratic algebra.