Non-commutative differential calculus on a Hopf algebra. An essential object, the representation \(\tau\) (Q1901186)

This paper is devoted to the study of covariant differential calculi on Hopf algebras (with not necessarily invertible antipode). After reviewing some basic facts on differential calculi over algebras, the author discusses covariant bimodules over Hopf algebras. She proves that left (right) covariant bimodules are in bijective correspondence with right (left) modules and characterizes bicovariant bimodules. Then, she switches to covariant differential calculi, characterizing them as quotients of the universal differential calculus and emphasizing the role of the right module structure on the space of left invariant forms (and vice versa). Finally, some examples and the relations to the commutative case are discussed.