Differential calculus on Hopf Group Coalgebra

Quantum groups, from a mathematical point of view, may be introduced by making emphasis on their q−deformed enveloping algebra aspects [1,2], which leads to the quantized enveloping algebras, or by making emphasis in the R−matrix formalism that describes the deformed group algebra. Also, they are mathematically well defined in the framework of Hopf algebra [3]. Quantum groups provide an interesting example of non-commutative geometry[4]. Non-commutative differential calculus on quantum groups is a fundamental tool needed for many applications [5,6]. S.L.Woronowicz [7] gave the general framework for bicovariant differential calculus on quantum groups following general ideas of A.Connes. Also, He showed that all important notions and formulae of classical Lie group theory admit a generalization to the quantum group case and he has restricted himself to compact matrix pseudogroups as introduced in [8].In contrast to the classical differential geometry on Lie groups, there is no functorial method to obtain a unique bicovariant differential calculus on a given quantum group [9].