Ore Extensions of Hopf Group Coalgebras

Abstract

Abstract. The aim of this paper is to generalize the theory of Hopf-Ore extension on Hopf algebras to Hopf group coalgebras. First the concept of Hopf-Ore extension of Hopf group coalgebra is introduced. Then we will give the necessary and sufficient condition for the Ore extensions to become a Hopf group coalgebra, and certain isomorphism between Ore extensions of Hopf group coalgebras are discussed.

The Ore Extension of Group-Cograded Hopf Coquasigroups

“…for any h ∈ H pq . The last equation coincides with (10). We now show that ( 13) and ( 14) imply ( 8) and (9).…”

“…Corollary 1. If group-cograded Hopf coquasigroup H = p∈G H p satisfies coassociativity, then the group-cograded Hopf coquasigroup-Ore extension is the Hopf coquasigroup-Ore extension in the sense of [10]. The main conclusion (i.e., Theorem 1) in this article is Theorem 3.5 in [10].…”

“…Jiao Zhengming [9] further generalized the Hopf-Ore extensions theory to Hopf coquasigroups. Afterwards, Wang Dingguo and Lu Daowei [10] extended the Ore extension of Hopf algebras to the Hopf group coalgebras.…”

Ore Extensions for the Sweedler’s Hopf Algebra H4

“…Panov [7] introduced the concept of Hopf-Ore extension R[x; σ, δ] of which the variable x is restricted to a skew primitive element and gave some equivalent descriptions. Later the Hopf-Ore extensions for some special Hopf algebras were obtained, such as quasitriangular Hopf algebras and multiplier Hopf algebras, see for example [8][9][10][11]. The authors [12] gave the realization of PBW-deformations of an quantum group via iterated Ore extensions.…”

Grothendieck Rings of 2$n^2$ dimensional Hopf Algebras $H_{2n^2}$

“…Later, Masuoka (see [14]) constructed this Hopf algebra as an extension of k[C 2 × C 2 ] by k[C 2 ]. Recently, using Ore extension(see [2,16,25,23,12,24,28]), an important method to constructing Hopf algebras, Pansera constructed an interesting class of semisimple Hopf algebras H 2n 2 in [17]. These Hopf algebras H 2n 2 of dimension 2n 2 are neither commutative nor cocommutative.…”