Universal Enveloping Algebras of PBW Type

TL;DR

This paper investigates the conditions under which universal enveloping algebras of PBW type exist, characterizing them as cosymmetric and applying this to recover classical PBW bases and classify braided bialgebra liftings.

Contribution

It provides a characterization of PBW type universal enveloping algebras as cosymmetric and links them to braided bialgebra liftings of Nichols algebras.

Findings

Universal enveloping algebra is of PBW type iff it is cosymmetric.

Recovered classical PBW bases for ordinary and restricted enveloping algebras.

Characterized braided bialgebra liftings of Nichols algebras as PBW type universal enveloping algebras.

Abstract

We continue our investigation of the general notion of universal enveloping algebra introduced in [A. Ardizzoni, \emph{A Milnor-Moore Type Theorem for Primitively Generated Braided Bialgebras}, J. Algebra \textbf{327} (2011), no. 1, 337--365]. Namely we study when such an algebra is of PBW type, meaning that a suitable PBW type theorem holds. We discuss the problem of finding a basis for a universal enveloping algebra of PBW type: As an application we recover the PBW basis both of an ordinary universal enveloping algebra and of a restricted enveloping algebra. We prove that a universal enveloping algebra is of PBW type if and only if it is cosymmetric. We characterize braided bialgebra liftings of Nichols algebras as universal enveloping algebras of PBW type.