ar X iv : m at h / 02 12 12 4 v 1 [ m at h . R A ] 9 D ec 2 00 2 COHOMOLOGY OF ABELIAN MATCHED PAIRS AND THE KAC SEQUENCE

The purpose of this paper is to introduce a cohomology theory for abelian matched pairs of Hopf algebras and to explore its relationship to Sweedler cohomology, to Singer cohomology and to extension theory. An exact sequence connecting these cohomology theories is obtained for a general abelian matched pair of Hopf algebras, generalizing those of Kac and Masuoka for matched pairs of finite groups and finite dimensional Lie algebras. The morphisms in the low degree part of this sequence are given explicitly, enabling concrete computations. In this paper we discuss various cohomology theories for Hopf algebras and their relation to extension theory. It is natural to think of building new algebraic objects from simpler structures , or to get information about the structure of complicated objects by decomposing them into simpler parts. Algebraic extension theories serve exactly that purpose, and the classification problem of such extensions is usually related to cohomology theories. In the case of Hopf algebras, extension theories are proving to be invaluable tools for the construction of new examples of Hopf algebras, as well as in the efforts to classify finite dimensional Hopf algebras. Hopf algebras, which occur for example as group algebras, as universal envelopes of Lie algebras, as algebras of representative functions on Lie groups, as coordinate algebras of algebraic groups and as Quantum groups, have many 'group like' properties. In particular, cocommutative Hopf algebras are group object in the category of cocommutative coalgebras, and are very much related to ordinary groups and Lie algebras. In fact, over an algebraically closed field of characteristic zero, such a Hopf algebra is a semi-direct product of a group algebra by a universal envelope of a Lie algebra, hence just a group algebra if finite dimensional (see [MM, Ca, Ko] for the connected case, [Gr1,2, Sw2] for the general case).