Hopf Algebras of Dimension 16

Abstract

We complete the classification of Hopf algebras of dimension 16 over an algebraically closed field of characteristic zero. We show that a non-semisimple Hopf algebra of dimension 16, has either the Chevalley property or its dual is pointed.

A Survey of Hopf Algebras of Low Dimension

“…This proposition has proved extremely useful in other attempts at classification, for example in [12,22,26]. A natural question is whether an analogue holds for Hopf algebras generated by simple subcoalgebras of dimension p, p an odd prime; see [5].…”

“…Two nonisomorphic nonpointed self-dual Hopf algebras of dimension 16 with coradical A 8 were described by Cȃlinescu, Dȃscȃlescu, Masuoka and Menini in [18] where the classification of Hopf algebras of dimension 16 with the Chevalley property is completed. Very recently García and Vay [26,Theorems 1.2,1.3] showed that the list of isomorphism classes above exhausts all possibilities for dimension 16, in other words, every Hopf algebra of dimension 16 has the Chevalley property or its dual is pointed.…”

Classifying Coalgebra Split Extensions of Hopf Algebras

“…A morphism ψ : A# f H → A# f H stabilizes A if and only if ψ(a#1 H ) = a# 1 H , for all a ∈ A. Taking into account the formula for ψ given by (14), we obtain that ψ stabilizes A if and only if u(a (1) )# p(a (2) ) = a# 1 H , i.e. u(a) = a and p(a) = ε A (a)1 H , for any a ∈ A.…”

“…On the other hand, there is no efficient cohomology theory for arbitrary Hopf algebras, similar to the one from group theory, to make a direct description of H 1 * (H, A) possible. One of the few examples known is [14,Lemma 2.8] where it is proved that any extension of H 4 by H 4 is equivalent to the trivial extension…”

On Hopf algebras of dimension 4p

“…By [HN,Lemma A.2], there exists an H -module isomorphism ϕ : P → S 2 P such that ϕ p n = id. It follows from [Ng4,Lemma 1.3] that Tr(ϕ) = 0.…”