Smash Coproduct Research Articles

A Morita-Takeuchi Context and Hopf Coquasigroup Galois Coextensions

For H a Hopf quasigroup and C, a left quasi H-comodule coalgebra, we show that the smash coproduct C⋊H (as a symmetry of smash product) is linked to some quotient coalgebra Q=C/CH*+ by a Morita-Takeuchi context (as a symmetry of Morita context). We use the Morita-Takeuchi setting to prove that for finite dimensional H, equivalent conditions for C/Q to be a Hopf quasigroup Galois coextension (as a symmetry of Galois extension). In particular, we consider a special case of quasigroup graded coalgebras as an application of our theory.

A (partial) weak smash coproduct

A (partial) weak smash coproduct

Some Results on Cohomology of Twisted Smash Coproduct

Some Results on Cohomology of Twisted Smash Coproduct

Two results on double crossed biproducts

Let HH be an algebra with a distinguished element εH∈H∗εH∈H∗ and C,DC,D two coalgebras. Based on the construction of Brzeziński’s crossed coproduct, under some suitable conditions, we introduce a coassociative coalgebra C×GTHβR×DC×TGHRβ×D which is a more general two-sided coproduct structure including two-sided smash coproduct. Necessary and sufficient conditions for C×GTHβR×DC×TGHRβ×D equipped with two-sided tensor product algebra C⊗H⊗DC⊗H⊗D to be a bialgebra (Hopf algebra) are provided. On the other hand, we obtain an improved version of the double crossed biproduct C⋆αHβ⋆DC⋆αHβ⋆D in [An extended form of Majid's double biproduct, J. Algebra Appl. 16 (4), 1760061, 2017] which induces a description of C⋆αHβ⋆DC⋆αHβ⋆D similar to Majid double biproduct C⋆H⋆DC⋆H⋆D and also present some related structures.

Double crossed biproducts and related structures

Let H be a bialgebra. Let be a linear map, where A is a left H-comodule coalgebra, and an algebra with a left H-weak action Let be a linear map, where B is a right H-comodule coalgebra, and an algebra with a right H-weak action In this paper, we improve the necessary conditions for the two-sided crossed product algebra and the two-sided smash coproduct coalgebra to form a bialgebra (called double crossed biproduct) such that the condition in Majid’s double biproduct (or double-bosonization) is one of the necessary conditions. On the other hand, we provide a more general two-sided crossed product algebra structure via Brzezński’s crossed product and give some applications.

Partial smash coproduct of multiplier Hopf algebras

In this work, we define partial (co)actions on multiplier Hopf algebras, we also present examples and properties. From a partial comodule coalgebra, we construct a partial smash coproduct generalizing the constructions made by L. Delvaux and E. Batista and J. Vercruysse.

Smash coproducts of monoidal comonads and Hom-entwining structures

Let $F$, $G$ be monoidal comonads on a monoidal category $\mathcal {C}$. The aim of this paper is to discuss the smash coproducts of $F$ and $G$. As an application, the smash coproduct of Hom-bialgebras is discussed. Further, the Hom-entwining structure and Hom-entwined modules are investigated.

On the cohomology of comodules over smash coproducts

We consider the category of comodules over a smash coproduct coalgebra [Formula: see text]. We show that there is a Grothendieck spectral sequence connecting the derived functors of the Hom functors coming from [Formula: see text]-colinear, [Formula: see text]-colinear and rational [Formula: see text]-colinear morphisms. We give several applications and connect our results to existing spectral sequences in the literature.

Duality theorem for L-R crossed coproducts

In this paper, the notion of L-R crossed coproduct is introduced as a unified approach for smash coproducts, crossed coproducts and L-R smash coproducts of Hopf algebras. A duality theorem for L-R crossed coproduct is proved.

Hom-L-R-smash biproduct and the category of Hom–Yetter–Drinfel’d–Long bimodules

Let [Formula: see text] be a Hom-bialgebra. In this paper, we firstly introduce the notion of Hom-L-R smash coproduct [Formula: see text], where [Formula: see text] is a Hom-coalgebra. Then for a Hom-algebra and Hom-coalgebra [Formula: see text], we introduce the notion of Hom-L-R-admissible pair [Formula: see text]. We prove that [Formula: see text] becomes a Hom-bialgebra under Hom-L-R smash product and Hom-L-R smash coproduct. Next, we will introduce a prebraided monoidal category [Formula: see text] of Hom–Yetter–Drinfel’d–Long bimodules and show that Hom-L-R-admissible pair [Formula: see text] actually corresponds to a bialgebra in the category [Formula: see text], when [Formula: see text] and [Formula: see text] are involutions. Finally, we prove that when [Formula: see text] is finite dimensional Hom-Hopf algebra, [Formula: see text] is isomorphic to the Yetter–Drinfel’d category [Formula: see text] as braid monoidal categories where [Formula: see text] is the tensor product Hom–Hopf algebra.

Semidirect products of weak multiplier Hopf algebras: Smash products and smash coproducts

ABSTRACTIn this paper, we will develop the smash product of weak multiplier Hopf algebras unifying the cases of Hopf algebras, weak Hopf algebras and multiplier Hopf algebras. We will show that the smash product R#A has a regular weak multiplier Hopf algebra structure if R and A are regular weak multiplier Hopf algebras. We shall investigate integrals on R#A. We also consider the result in the ∗-situation and new examples. Dually, we consider the smash coproduct of weak multiplier Hopf algebras under an appropriate form and integrals on the smash coproduct and we obtain results in the ∗-situation.

Tensor Products and Support Varieties for Some Noncocommutative Hopf Algebras

We explore questions of projectivity and tensor products of modules for finite dimensional Hopf algebras. We construct many classes of examples in which tensor powers of nonprojective modules are projective and tensor products of modules in one order are projective but in the other order are not. Our examples are smash coproducts with duals of group algebras, some having algebra and coalgebra structures twisted by cocycles. We apply support variety theory for these Hopf algebras as a tool in our investigations.

An extended form of Majid’s double biproduct

Let [Formula: see text] be a bialgebra. Let [Formula: see text] be a linear map, where [Formula: see text] is a left [Formula: see text]-module algebra, and a coalgebra with a left [Formula: see text]-weak coaction. Let [Formula: see text] be a linear map, where [Formula: see text] is a right [Formula: see text]-module algebra, and a coalgebra with a right [Formula: see text]-weak coaction. In this paper, we extend the construction of two-sided smash coproduct to two-sided crossed coproduct [Formula: see text]. Then we derive the necessary and sufficient conditions for two-sided smash product algebra [Formula: see text] and [Formula: see text] to be a bialgebra, which generalizes the Majid’s double biproduct in [Double-bosonization of braided groups and the construction of [Formula: see text], Math. Proc. Camb. Philos. Soc. 125(1) (1999) 151–192] and the Wang–Wang–Yao’s crossed coproduct in [Hopf algebra structure over crossed coproducts, Southeast Asian Bull. Math. 24(1) (2000) 105–113].

Twisted tensor biproduct monoidal Hom–Hopf algebras

Let [Formula: see text] be a monoidal Hom-bialgebra, [Formula: see text] a monoidal Hom-algebra and a monoidal Hom-coalgebra. Let [Formula: see text] and [Formula: see text] be two linear maps. First, we construct the [Formula: see text]-smash product monoidal Hom-algebra [Formula: see text] and [Formula: see text]-smash coproduct monoidal Hom-coalgebra [Formula: see text]. Second, the necessary and sufficient conditions for [Formula: see text] and [Formula: see text] to be a monoidal Hom-bialgebra are obtained, which generalizes the results in [8, 11]. Lastly, we give some examples and applications.

Smash积与Smash余积的对偶性

本文探究了模代数与模余代数、余模代数与余模余代数之间的相互关系，并从含于余代数A0内的余模余代数A□出发，进一步刻画了Smash积与Smash余积的对偶性。 This paper mainly discusses the relations between module algebra and module coalgebra, comod-ule algebra and comodule coalgebra. Last, from the comodule coalgebra A□ contained in A0, we further characterize duality between the smash product and smash coproduct.

Double biproduct Hom-bialgebra and related quasitriangular structures

Let (H, β) be a Hom-bialgebra such that β2 = idH. (A, αA) is a Hom-bialgebra in the left-left Hom-Yetter-Drinfeld category HHYD and (B, αB) is a Hom-bialgebra in the right-right Hom-Yetter-Drinfeld category YDHH. The authors define the two-sided smash product Hom-algebra \(\left( {A\natural H\natural B,{\alpha _A} \otimes \beta \otimes {\alpha _B}} \right)\) and the two-sided smash coproduct Homcoalgebra \(\left( {A\diamondsuit H\diamondsuit B,{\alpha _A} \otimes \beta \otimes {\alpha _B}} \right)\). Then the necessary and sufficient conditions for \(\left( {A\natural H\natural B,{\alpha _A} \otimes \beta \otimes {\alpha _B}} \right)\) and \(\left( {A\diamondsuit H\diamondsuit B,{\alpha _A} \otimes \beta \otimes {\alpha _B}} \right)\) to be a Hom-bialgebra (called the double biproduct Hom-bialgebra and denoted by \(\left( {A_\diamondsuit ^\natural H_\diamondsuit ^\natural B,{\alpha _A} \otimes \beta \otimes {\alpha _B})} \right)\) are derived. On the other hand, the necessary and sufficient conditions for the smash coproduct Hom-Hopf algebra \(\left( {A\diamondsuit H,{\alpha _A} \otimes \beta } \right)\) to be quasitriangular are given.

Hopf automorphisms and twisted extensions

We give some applications of a Hopf algebra constructed from a group acting on another Hopf algebra [Formula: see text] as Hopf automorphisms, namely Molnar’s smash coproduct Hopf algebra. We find connections between the exponent and Frobenius–Schur indicators of a smash coproduct and the twisted exponents and twisted Frobenius–Schur indicators of the original Hopf algebra [Formula: see text]. We study the category of modules of the smash coproduct.

Quotients and Hopf images of a smash coproduct

We describe the Hopf algebra quotients and Hopf images of the smash coproduct of a group algebra by the algebra of functions on a finite group.

On Radford Biproduct

Let H be a bialgebra, A an algebra, and a left H-comodule coalgebra. Let f: H ⊗ H → A ⊗ H and R: H ⊗ A → A ⊗ H be two linear maps. Let B be a right H-module algebra and also a right H-comodule coalgebra. In this paper, necessary and sufficient conditions are given for the one-sided Brzeziński's crossed product algebra and the two-sided smash coproduct coalgebra A × H × B to form a bialgebra, which we call the Brzeziński's double biproduct. The celebrated Radford biproduct [18], Majid's double biproduct [13], Agore and Militaru's unified product [2], and the Wang–Jiao–Zhao's crossed product [21] are all derived as special cases. On the other hand, we construct a class of Radford biproducts by a twisting method.

Two Interacting Coordinate Hopf Algebras of Affine Groups of Formal Series on a Category

A locally finite category is defined as a category in which every arrow admits only finitely many different ways to be factorized by composable arrows. The large algebra of such categories over some fields may be defined, and with it a group of invertible series (under multiplication). For certain particular locally finite categories, a substitution operation, generalizing the usual substitution of formal power series, may be defined, and with it a group of reversible series (invertible under substitution). Moreover, both groups are actually affine groups. In this contribution, we introduce their coordinate Hopf algebras which are both free as commutative algebras. The semidirect product structure obtained from the action of reversible series on invertible series by anti-automorphisms gives rise to an interaction at the level of their coordinate Hopf algebras under the form of a smash coproduct.