We prove that for a specific class of Lie bialgebras, there exists a natural differential calculus. This class consists of the Lie bialgebras for which the dual Lie bialgebra is of triangular type. The differential calculus is explicitly constructed with the help of the $R$-matrix from the dual. The method is illustrated by several examples.

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). A natural differential calculus on Lie bialgebras with dual of triangular type. Department of Analysis, Algebra and Geometry [AM]. CWI.