Universal Enveloping Algebras of PBW Type

Abstract

Abstract We continue our investigation of the general notion of universal enveloping algebra introduced in [A. Ardizzoni, A Milnor–Moore type theorem for primitively generated braided Bialgebras, J. Algebra 327(1) (2011), 337–365]. Namely, we study a universal enveloping algebra when it is of Poincaré–Birkhoff–Witt (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 characterise braided bialgebra liftings of Nichols algebras as universal enveloping algebras of PBW type.