Twistings, crossed coproducts, and Hopf‐Galois coextensions

Abstract

Let H be a Hopf algebra. Ju and Cai (2000) introduced the notion of twisting of an H-module coalgebra. In this paper, we study the relationship between twistings, crossed coproducts, and Hopf-Galois coextensions. In particular, we show that a twisting of an H-Galois coextension remains H-Galois if the twisting is invertible

Two results on double crossed biproducts

“…0 0 0 u (1) 0 u (1) 0 0 u (2) 0 0 u (2) 0 u (3) 0 0 0 u (3) , its coalgebra structure is defined by ∆(u (0) ) = u (0) ⊗ u (0) + u (1) ⊗ u (3) + u (3) ⊗ u (1) + u (2) ⊗ u (2) ; ∆(u (1) ) u (0) ⊗ u (1) + u (1) ⊗ u (0) + u (2) ⊗ u (3) + u (3) ⊗ u (2) ; ∆(u (2) ) = u (0) ⊗ u (2) + u (2) ⊗ u (0) + u (1) ⊗ u (1) + u (3) ⊗ u (3) ; ∆(u (3) ) = u (0) ⊗ u (3) + u (3) ⊗ u (0) + u (1) ⊗ u (2) + u (2) ⊗ u (1) .…”

“…, ρ(u (1) ) := λ ⊗ u (1) + ς ⊗ u (3) , ρ(u (2) ) := λ ⊗ u (2) + ς ⊗ u (2) , ρ(u (3) ) := λ ⊗ u (3) + ς ⊗ u (1) ; (1) ) := ς ⊗ ς, α(u (2) ) := 0, α(u (3) ) := 0; (1) ) := ς ⊗ ς, β(v (2) ) := 0, β(v (3) ) := 0. (1) , c 7 = u (1) ⊗ λ ⊗ v (2) , c 8 = u (1) ⊗ λ ⊗ v (3) , c 9 = u (2) ⊗ λ ⊗ v (0) , c 10 = u (2) ⊗ λ ⊗ v (1) , c 11 = u (2) ⊗ λ ⊗ v (2) , c 12 = u (2) ⊗ λ ⊗ v (3) , c 13 = u (3) ⊗ λ ⊗ v (0) , c 14 = u (3) ⊗ λ ⊗ v (1) , c 15 = u (3) ⊗ λ ⊗ v (2) , c 16 = u (3) ⊗ λ ⊗ v (3) , c 17 = u (0) ⊗ ς ⊗ v (0) , c 18 = u (0) ⊗ ς ⊗ v (1) , c 19 = u (0) ⊗ ς ⊗ v (2) , c 20 = u (0) ⊗ ς ⊗ v (3) , c 21 = u (1) ⊗ ς ⊗ v (0) , c 22 = u (1) ⊗ ς ⊗ v (1) , c 23 = u (1) ⊗ ς ⊗ v (2) , c 24 = u (1) ⊗ ς ⊗ v (3) , c 25 = u (2) ⊗ ς ⊗ v (0) , c 26 = u (2) ⊗ ς...…”

“…More generally, if H is a vector space with a distinguished element ε H ∈ H * and C is a coalgebra, then Brzeziński provided a coassociative coalgebra C × G T H (here we call it left Brzeziński crossed coproduct) under some conditions [1]. For the research of crossed coproduct, see [2,3,5,8,11,14].…”

Some interactions between Hopf Galois extensions and noncommutative rings

“…• Hopf Galois coextensions: We studied above the relation between HGE and the normal basis property; however, there exists a coalgebra version of the normal basis property involving the notion of crossed coproduct and cleft coextension, introduced by [31]. This is further studied in [21] addressing the notion of Hopf Galois coextension and twisting techniques. More recently, this theory was used in [55] to show that Hopf Galois coextensions of coalgebras are the sources of stable anti Yetter-Drinfeld modules.…”

On representation theory of total (co)integrals

“…This is because can(c ⊗ 1 H ) and can ′ : [CDY,Lemma 4.1] are both injective. This map is not necessarily surjective in general.…”