We discuss a method to construct a De Rham complex (differential algebra) of Poincar'e-Birkhoff-Witt-type on the universal enveloping algebra of a Lie algebra $g$. We determine the cases in which this gives rise to a differential Hopf algebra that naturally extends the Hopf algebra structure of $U(g)$. The construction of such differential structures is interpreted in terms of colour Lie superalgebras.

Universal enveloping algebras of Lie algebras (msc 16S30), Rings and algebras with additional structure (msc 16Wxx), Universal enveloping (super)algebras (msc 17B35), Quantum groups (quantized enveloping algebras) and related deformations (msc 17B37), Quantum groups and related algebraic methods (msc 81R50)
CWI
Department of Analysis, Algebra and Geometry [AM]

van den Hijligenberg, N.W, & Martini, R. (1995). Differential Hopf algebra structures on the universal enveloping algebra ofa Lie algebra. Department of Analysis, Algebra and Geometry [AM]. CWI.