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

2018 ◽  
Vol 17 (07) ◽  
pp. 1850133 ◽  
Author(s):  
Daowei Lu ◽  
Xiaohui Zhang
Keyword(s):  
Hopf Algebra ◽  
Tensor Product ◽  
Monoidal Category ◽  
Admissible Pair ◽  
Smash Product ◽  
Monoidal Categories ◽  
Finite Dimensional

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.

On the Hopf algebra dual of an enveloping algebra

1982 ◽  
Vol 91 (2) ◽  
pp. 215-224 ◽  
Author(s):  
Stephen Donkin
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Algebraic Groups ◽  
Field Extension ◽  
Smash Product ◽  
Enveloping Algebras ◽  
Enveloping Algebra ◽  
Finite Dimensional ◽  
Polycyclic Groups ◽  
Algebra Over A Field

In (1) it is claimed that the main results of that paper have applications to the representation theory of algebraic groups, of polycyclic groups and of Lie algebras. An application to algebraic groups is given in Corollary 6·4 of (1), the applications to polycyclic groups are given in (2), the purpose of this work is to deal with the outstanding case of enveloping algebras. To make use of the results of (1), in this context, we show that the Hopf algebra dual of the enveloping algebra of a finite dimensional Lie algebra over a field of characteristic zero is quasi-affine (see § 1·5). This is done by an easy field extension argument and a generalization, to the Hopf algebra dual of the smash product of Hopf algebras, of Proposition 1·6·3 of (2) on the dual of the group algebra of a semidirect product of groups. Since this paper is aimed at those readers interested in enveloping algebras, the Hopf theoretic aspects are dealt with at a fairly leisurely pace.

scholarly journals Comodules over weak multiplier bialgebras

2014 ◽  
Vol 25 (05) ◽  
pp. 1450037 ◽  
Author(s):  
Gabriella Böhm
Keyword(s):  
Tensor Product ◽  
Hopf Algebras ◽  
Monoidal Category ◽  
Base Algebra ◽  
Finite Dimensional ◽  
Linear Maps ◽  
Multiplier Hopf Algebras ◽  
Hopf Modules ◽  
Total Algebra ◽  
Module Structures

This is a sequel paper of [Weak multiplier bialgebras, Trans. Amer. Math. Soc., in press] in which we study the comodules over a regular weak multiplier bialgebra over a field, with a full comultiplication. Replacing the usual notion of coassociative coaction over a (weak) bialgebra, a comodule is defined via a pair of compatible linear maps. Both the total algebra and the base (co)algebra of a regular weak multiplier bialgebra with a full comultiplication are shown to carry comodule structures. Kahng and Van Daele's integrals [The Larson–Sweedler theorem for weak multiplier Hopf algebras, in preparation] are interpreted as comodule maps from the total to the base algebra. Generalizing the counitality of a comodule to the multiplier setting, we consider the particular class of so-called full comodules. They are shown to carry bi(co)module structures over the base (co)algebra and constitute a monoidal category via the (co)module tensor product over the base (co)algebra. If a regular weak multiplier bialgebra with a full comultiplication possesses an antipode, then finite-dimensional full comodules are shown to possess duals in the monoidal category of full comodules. Hopf modules are introduced over regular weak multiplier bialgebras with a full comultiplication. Whenever there is an antipode, the Fundamental Theorem of Hopf Modules is proven. It asserts that the category of Hopf modules is equivalent to the category of firm modules over the base algebra.

scholarly journals On Hopf DeMeyer-Kanzaki Galois extensions

2003 ◽  
Vol 2003 (26) ◽  
pp. 1627-1632
Author(s):  
George Szeto ◽  
Lianyong Xue
Keyword(s):  
Hopf Algebra ◽  
Galois Extension ◽  
Smash Product ◽  
Module Algebra ◽  
Galois Extensions ◽  
Finite Dimensional ◽  
Dual Hopf Algebra ◽  
Finite Dimensional Hopf Algebra ◽  
Algebra Over A Field

LetHbe a finite-dimensional Hopf algebra over a fieldk,Ba leftH-module algebra, andH∗the dual Hopf algebra ofH. For anH∗-Azumaya Galois extensionBwith centerC, it is shown thatBis anH∗-DeMeyer-Kanzaki Galois extension if and only ifCis a maximal commutative separable subalgebra of the smash productB#H. Moreover, the characterization of a commutative Galois algebra as given by S. Ikehata (1981) is generalized.

scholarly journals Separable Functors and Formal Smoothness

2007 ◽  
Vol 1 (3) ◽  
pp. 535-582 ◽  
Author(s):  
Alessandro Ardizzoni
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Algebra Homomorphism ◽  
Monoidal Categories ◽  
Invariant Integrals ◽  
Finite Dimensional ◽  
Finite Dimensional Case ◽  
Algebra Theory ◽  
Formal Smoothness ◽  
Dual Case

AbstractThe natural problem we approach in the present paper is to show how the notion of formally smooth (co)algebra inside monoidal categories can substitute that of (co)separable (co)algebra in the study of splitting bialgebra homomorphisms. This is performed investigating the relation between formal smoothness and separability of certain functors and led to other results related to Hopf algebra theory. Between them we prove that the existence of ad-(co)invariant integrals for a Hopf algebra H is equivalent to the separability of some forgetful functors. In the finite dimensional case, this is also equivalent to the separability of the Drinfeld Double D(H) over H. Hopf algebras which are formally smooth as (co)algebras are characterized. We prove that if π : E → H is a bialgebra surjection with nilpotent kernel such that H is a Hopf algebra which is formally smooth as a K-algebra, then π has a section which is a right H-colinear algebra homomorphism. Moreover, if H is also endowed with an ad-invariant integral, then this section can be chosen to be H-bicolinear. We also deal with the dual case.

THE CALABI–YAU PROPERTY OF TWISTED SMASH PRODUCTS

2013 ◽  
Vol 13 (03) ◽  
pp. 1350118 ◽  
Author(s):  
DINGGUO WANG ◽  
YUANYUAN KE
Keyword(s):  
Hopf Algebra ◽  
Smash Product ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Semisimple Hopf Algebra ◽  
Finite Dimensional ◽  
Necessary And Sufficient

Let H be a finite-dimensional cocommutative semisimple Hopf algebra and A * H a twisted smash product. The Calabi–Yau (CY) property of twisted smash product is discussed. It is shown that if A is a CY algebra of dimension dA, a necessary and sufficient condition for A * H to be a CY Hopf algebra is given.

Grothendieck Groups of a Class of Quantum Doubles

2008 ◽  
Vol 15 (03) ◽  
pp. 431-448 ◽  
Author(s):  
Yun Zhang ◽  
Feng Wu ◽  
Ling Liu ◽  
Hui-Xiang Chen
Keyword(s):  
Hopf Algebra ◽  
Tensor Product ◽  
Grothendieck Group ◽  
Ring Structure ◽  
Quantum Double ◽  
Simple Modules ◽  
Loewy Length ◽  
Finite Dimensional ◽  
Grothendieck Groups

Let k be a field and An(ω) be the Taft n2-dimensional Hopf algebra. When n is odd, the Drinfeld quantum double D(An(ω)) of An(ω) is a ribbon Hopf algebra. We have constructed an n4-dimensional Hopf algebra Hn(p,q) which is isomorphic to D(An(ω)) if p ≠ 0 and q = ω-1, and studied the irreducible and finite-dimensional representations of Hn(1,q). In this paper, we continue our study of Hn(1,q), examine the Grothendieck group G0(Hn(1,q)) ≅ G0(D(An(ω)), and describe its ring structure. We also give the Loewy length of the tensor product of two simple modules over Hn(1,q).

scholarly journals Mixed vs Stable Anti-Yetter-Drinfeld Contramodules

2021 ◽  
Author(s):  
Ilya Shapiro ◽  
Keyword(s):  
Hopf Algebra ◽  
Monoidal Category ◽  
Cyclic Homology ◽  
Mixed Complexes ◽  
Finite Dimensional ◽  
Category Of Modules ◽  
Sweedler’S Hopf Algebra ◽  
Finite Dimensional Hopf Algebra

We examine the cyclic homology of the monoidal category of modules over a finite dimensional Hopf algebra, motivated by the need to demonstrate that there is a difference between the recently introduced mixed anti-Yetter-Drinfeld contramodules and the usual stable anti-Yetter-Drinfeld contramodules. Namely, we show that Sweedler's Hopf algebra provides an example where mixed complexes in the category of stable anti-Yetter-Drinfeld contramodules (previously studied) are not equivalent, as differential graded categories to the category of mixed anti-Yetter-Drinfeld contramodules (recently introduced).

scholarly journals On Hopf Galois Hirata extensions

2003 ◽  
Vol 2003 (64) ◽  
pp. 4033-4039
Author(s):  
George Szeto ◽  
Lianyong Xue
Keyword(s):  
Hopf Algebra ◽  
Smash Product ◽  
Sufficient Condition ◽  
Separable Extension ◽  
Finite Dimensional ◽  
Dual Hopf Algebra ◽  
Finite Dimensional Hopf Algebra ◽  
Algebra Over A Field

LetHbe a finite-dimensional Hopf algebra over a fieldK,H*the dual Hopf algebra ofH, andBa rightH*-Galois and Hirata separable extension ofBH. ThenBis characterized in terms of the commutator subringVB(BH)ofBHinBand the smash productVB(BH)#H. A sufficient condition is also given forBto be anH*-Galois Azumaya extension ofBH.

scholarly journals Nuclear subalgebras of UHF C*-algebras

1986 ◽  
Vol 29 (1) ◽  
pp. 97-100 ◽  
Author(s):  
R. J. Archbold ◽  
Alexander Kumjian
Keyword(s):  
Tensor Product ◽  
Inductive Limit ◽  
Inductive Limits ◽  
C Algebra ◽  
C Algebras ◽  
Finite Dimensional ◽  
The Family ◽  
Algebraic Tensor Product

A C*-algebra A is said to be approximately finite dimensional (AF) if it is the inductive limit of a sequence of finite dimensional C*-algebras(see [2], [5]). It is said to be nuclear if, for each C*-algebra B, there is a unique C*-norm on the *-algebraic tensor product A ⊗B [11]. Since finite dimensional C*-algebras are nuclear, and inductive limits of nuclear C*-algebras are nuclear [16];,every AF C*-algebra is nuclear. The family of nuclear C*-algebras is a large and well-behaved class (see [12]). The AF C*-algebras for a particularly tractable sub-class which has been completely classified in terms of the invariant K0 [7], [5].

Invariants of 3-manifolds derived from universal invariants of framed links

1995 ◽  
Vol 117 (2) ◽  
pp. 259-273 ◽  
Author(s):  
Tomotada Ohtsuki
Keyword(s):  
Hopf Algebra ◽  
Quantum Group ◽  
Independent Method ◽  
Topological Invariant ◽  
Finite Dimensional

Reshetikhin and Turaev [10] gave a method to construct a topological invariant of compact oriented 3-manifolds from a ribbon Hopf algebra (e.g. a quantum group Uq(sl2)) using finite-dimensional representations of it. In this paper we give another independent method to construct a topological invariant of compact oriented 3-manifolds from a ribbon Hopf algebra via universal invariants of framed links without using representations of the algebra. For Uq(sl2) these two methods give different invariants of 3-manifolds.

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