Low Dimensional Cocommutative Connected Hopf Algebras

William M. Singer’s theory of extensions of connected Hopf algebras is used to give a complete list of the cocommutative connected Hopf algebras over a field of positive characteristic p which have vector space dimension less than or equal to p3. The theory shows that there are exactly two noncommutative non-primitively generated Hopf algebras on the list, one of which is the Hopf algebra corresponding to the sub-Hopf algebra of the Steenrod algebra generated by P 1 and P p. The commutative Hopf algebras are found using Borel’s theorem and the primitively generated Hopf algebras using restricted Lie algebras. Introduction. In this paper we study low dimensional cocommutative connected k-Hopf algebras for k a field of positive characteristic p using William M. Singer’s theory of extensions of connected Hopf algebras [2]. Specifically, any finite dimensional cocommutative connected k-Hopf algebra occurs in a central extension A → C → B where B is a cocommutative connected k-Hopf algebra of vector space dimension strictly less than that of C and A is polynomial on one generator truncated at height p or exterior on one generator. Singer describes a cohomology group H(B,A) which classifies such extensions up to equivalence [2,proposition 5.1]. We use this group and induction on the dimension of C to give a complete list of the cocommutative connected k-Hopf algebras of vector space dimension less than or equal to p. A list of the commutative connected k-Hopf algebras of dimension less than or equal to p could also be obtained by taking duals. In practice it can be difficult to compute Singer’s cohomology group, or even to calculate the Hopf algebra structure on C determined by an element in that group. Fortunately, we can avoid these computations for all but a few specific cases. When C is commutative and k is a perfect field, the algebra structure is determined by Borel’s theorem [1,theorem 7.11], and the possible coproducts are easily deduced when the dimension of C is small. If C is noncommutative but primitively generated, then it is the universal enveloping algebra of a nonabelian connected restricted Lie algebra [1, theorem 6.11]. When the dimension of C is small, it is not difficult to give a list of these Lie algebras. We use Singer’s theory for the remaining cases : C noncommutative and non-primitively generated or C commutative and k not a perfect field. There are relatively few of these. Since Singer’s theory is not widely known, we give a brief summary in section one. A deep understanding of his results is not necessary for our purposes, but we will use his terminology and his definition of the group classifying extensions. In section two we study central extensions A → C → B with A polynomial in one variable truncated at height p or exterior on one generator and with B and C cocommutative connected k-Hopf algebras. We are able to characterize those elements of H(B,A) which determine extensions with C commutative or with C primitively generated. We also construct a small piece of an exact sequence which is helpful in calculating H(B,A). In section three we apply these results to low dimensional cocommutative connected k-Hopf algebras. We show that there are only two pairs (A,B) which can give a C of dimension less than or equal to p when C is noncommutative and non-primitively generated. We calculate H(B,A) in these cases and show that there are only two such Hopf algebras. Borel’s theorem, supplemented by the theory of section two when k is not perfect, and restricted Lie algebras are used to complete the list. 1991 Mathematics Subject Classification. Primary 16W30 57T05 ; Secondary 18G60.