Classifying Coalgebra Split Extensions of Hopf Algebras

Abstract

For a given Hopf algebra A we classify all Hopf algebras E that are coalgebra split extensions of A by H4, where H4is the Sweedler's four-dimensional Hopf algebra. Equivalently, we classify all crossed products of Hopf algebras A# H4by computing explicitly two classifying objects: the cohomological "group" [Formula: see text] and CRP (H4, A) ≔ the set of types of isomorphisms of all crossed products A# H4. All crossed products A# H4are described by generators and relations and classified: they are parameterized by the set [Formula: see text] of all central primitive elements of A. Several examples are worked out in detail: in particular, over a field of characteristic p ≥ 3 an infinite family of non-isomorphic Hopf algebras of dimension 4p is constructed. The groups of automorphisms of these Hopf algebras are also described.

The Classification of All Crossed Products H₄#k[C_n]

“…From now on k will be an arbitrary field and we shall use ⊗ instead of ⊗ k . For the comultiplication of a Hopf algebra we use Sweedler's Σ-notation with suppressed summation sign: ∆(a) = a (1) ⊗a (2) . If A and H are Hopf algebras and f :…”

“…Two coalgebra split extensions (E, i, π), (E , i , π ) of A by H are called equivalent if there exists an isomorphism of Hopf algebras ψ : E → E that stabilizes A and co-stabilizes H, i.e. the following diagram H → E is a unit preserving coalgebra map that splits π then the action = ϕ and the cocycle f = f ϕ implemented by ϕ are given by: h a := ϕ(h (1) )aSϕ(h (2) ) and f (g, h) := ϕ(g (1) )ϕ(h (1) )Sϕ(g (2)…”

“…Crp(H, A). As in the case of groups, it turns out that the two classifying objects H 2 (H, A) and Crp(H, A) are different: [2,Proposition 4.2] proves that for the pair of Hopf algebras (H,…”

“…Let (H 4 , k[C n ], , f t ) and (H 4 , k[C n ], , f t ) be the crossed systems corresponding to (t, λ) and respectively (t , λ ) given in Theorem 3. (CP1) u(a (1) ) ⊗ p(a (2) ) = u(a (2) ) ⊗ p(a (1) ), (CP2) r(h (1) ) ⊗ v(h (2) ) = r(h (2) ) ⊗ v(h (1) ), (CP3) u(ab) = u(a (1) ) p(a (2) ) u(b (1) ) f p(a (3) ), p(b (2) ) , (1) , g (1) ) v(h (2) g (2) ),…”

“…h(2) ), for all g, h ∈ H and a ∈ A. Thus, the classification of all coalgebra split extensions of A by H reduces to the classification of all crossed products A# f H. The classifying object for all coalgebra split extensions of A by H, denoted by H 2(H, A), was introduced in [2, Remark 2.4].…”