Cohomological Properties of Differential Calculi on Hopf Algebras

In the approach to the differential calculus on quantum groups proposed in [1], besides the obvious notion of differential d, the theory is founded on the notion of bicovariant bimodule; the algebraic nature of this construction extends the properties of differential forms to the noncommutative situation. Following this idea a general treatment and a classification of differential calculi have been constructed for quantum groups obtained as deformation of semisimple Lie groups making use of the quasitriangularity property [2,3]. In this report we give an intrinsic treatment of the results we developed in [4] connecting the differential calculi on Hopf algebras to the Drinfeld double [5]. In the first place we recover that bicovariant bimodules are in one to one correspondence with the Drinfeld double representations [6]; we then introduce a Hochschild cohomology of the algebra of functions and discuss the main result stating that each differential calculus is associated to a 1-cocycle satisfying an additional invariance condition with respect to a natural action [4]. Defining a Hochschild cohomology of the double, the above invariance becomes a condition with respect to the enveloping algebra component of the double that must be added to the 1-cocycle relation. The general classification of differential calculi is therefore reduced to a cohomological problem, which can be performed with the more usual and efficient tools. Moreover a supply of differential calculi is obtained by observing that the coboundary operator maps invariant 0-cochains into invariant 1-coboundaries. The construction we present is completely independent of the quasi-triangular property of Hopf algebras and can obviously be applied to classical groups, both Lie and discrete or finite. An interesting feature is that all the known differential calculi on quantum and finite groups correspond to coboundaries, at difference with the usual Lie group case, in which no invariant coboundary exists and the classical differential calculus is determined by a nontrivial 1-cocycle.