Quantum groups and quantum algebras as well as non-commutative differential geometry are important in mathematics. They are also considered useful tools for model building in statistical and quantum physics. This book, addressing scientists and postgraduates, contains a detailed and rather complete presentation of the algebraic framework. Introductory chapters deal with background material such as Lie and Hopf superalgebras, Lie super-bialgebras, or formal power series. A more general approach to differential forms, and a systematic treatment of cyclic and Hochschild cohomologies within their universal differential envelopes are developed. Quantum groups and quantum algebras are treated extensively. Great care was taken to present a reliable collection of formulae and to unify the notation, making this volume a useful work of reference for mathematicians and mathematical physicists.

Lie Algebras.- Lie Superalgebras.- Coalgebras and Z2-Graded Hopf Algebras.- Formal Power Series with Homogeneous Relations.- Z2-Graded Lie-Cartan Pairs.- Real Lie-Hopf Superalgebras.- Universal Differential Envelope.- Quantum Groups.- Categorial Viewpoint.