Hopf Algebra Extensions of Monogenic Hopf Algebras

William M. Singer has described a cohomology theory of connected Hopf algebras which classifies extensions of a cocommutative Hopf algebra by a commutative Hopf algebra in much the same way as the cohomology of groups classifies extensions of a group by an abelian group. We compute these cohomology groups for monogenic Hopf algebras, construct an action of the base ring on the cohomology groups in the case of trivial matched pairs, and use these results to further study Singer’s cohomology. Introduction. Extensions of Hopf algebras are an important tool for studying Hopf algebras and their cohomologies, but the theory of such extensions does not seem to be widely known. The connected case was described by William M. Singer [11] in 1972 and is a generalization of earlier work of V.K.A.M. Gugenheim [3] on central extensions of Hopf algebras. The theory for general (ungraded or non-connected) Hopf algebras is a special case of work by Pachuashvili, published in Russian in 1985 and in English in [9], and is described explicitly by Hofstetter [5]. Singer’s theory is essentially a self-dual version of the classical extension theory of groups and is accessible to anyone who is familiar with extensions of groups and the basics of Hopf algebras. Cohomology groups are defined such that H(B,A) classifies all extensions of B by A up to equivalence. But in spite of many similarities to group theory, Singer’s cohomology groups appear quite difficult to compute the results presented here are the only computations of which we are aware. In the first part of this paper we determine Singer’s cohomology groups for the monogenic Hopf algebras over a base ring R and their duals. For the tensor algebra T (x) and its dual, the shuffle algebra Γ(x), we are able to compute all of the groups H(T (x), A) and H(B,Γ(x)). For the exterior algebra Λ(x) over a 2divisible ring we obtain spectral sequences which compute H(Λ(x), A) and H(B,Λ(x)). For the truncated tensor algebras Tn(x) for n = p l or n = 2p and their duals Γn(x), we compute H (Tn(x), A), H (Tn(x), A), H(B,Γn(x)) and H (B,Γn(x)). The results for the monogenic cases are given in terms of Cotor ∗,∗ A (R,R) and the results for the duals are in terms of Ext∗,∗ B (R,R), the usual cohomology of B. Since every nontrivial finite dimensional cocommutative connected Hopf algebra over a field k of characteristic p has a central sub-Hopf algebra of the form Tp(x) or Λ(x) (see, for example [12,proposition 1.2]), the results of part one can be used give a list of the low dimensional cocommutative connected Hopf algebras over a field of positive characteristic. An attempt to do this without actually computing Singer’s cohomology group can be found in [4] and suggested the methods used here. In the second part we consider “trivial coefficients” (trivial matched pairs) and describe actions of the base ring R on Singer’s cohomology which have natural interpretations in terms of extensions. These actions give the cohomology the structure of an abelian group with multiplicative R-action and are easily described in the cases covered in part one. Finally, we apply these computations to study Singer’s cohomology itself. We are able to show that there is in general no long exact sequence for extensions in either variable and to calculate the “acyclics” for H∗(B,−) (Hopf algebras B for which Hn(B,−) is trivial for n > 2) when R is a 2-divisible ring. Singer’s Theory of Extensions of Connected Hopf Algebras. In the classical theory of extensions of groups by abelian groups, one shows that each extension A→ E → G determines an action σ of G on A by conjugation and a twisting function τ : G×G→ A which encode the information about products in E. In particular, the extension is equivalent to A→ A×G→ G if A×G has the twisted product determined by 1991 Mathematics Subject Classification. Primary 16W30 57T05 ; Secondary 18G60.