Ideals of Finite-Dimensional Pointed Hopf Algebras of Rank One

Let H be a finite-dimensional pointed Hopf algebra of rank one over an algebraically closed field of characteristic zero.In this paper we show that any finite-dimensional indecomposable H-module is generated by one element.In particular,any indecomposable submodule of H under the adjoint action is generated by a special element of H.Using this result,we show that the Hopf algebra H is a principal ideal ring,i.e.,any two-sided ideal of H is generated by one element.As an application,we give explicitly the generators of ideals,primitive ideals,maximal ideals and completely prime ideals of the Taft algebras.     

Invariants of Spin Three-Manifolds From Chern-Simons Theory and Finite-Dimensional Hopf Algebras

A version of Kirby calculus for spin and framed three-manifolds is given and is used to construct invariants of spin and framed three-manifolds in two situations. The first is ribbon *-categories which possess odd degenerate objects. This case includes the quantum group situations corresponding to the half-integer level Chern-Simons theories conjectured to give spin TQFTs by Dijkgraaf and Witten (1990, Commun. Math. Phys.129, 393-429). In particular, the spin invariants constructed by Kirby and Melvin (1991, Invent. Math.105, 473-545) are shown to be identical to the invariants associated to SO(3). Second, an invariant of spin manifolds analogous to the Hennings invariant is constructed beginning with an arbitrary factorizable, unimodular quasitriangular Hopf algebra. In particular a framed manifold invariant is associated to every finite-dimensional Hopf algebra via its quantum double, and is conjectured to be identical to Kuperberg's noninvolutory invariant of framed manifolds associated to that Hopf algebra.     

A Hopf Operad of Forests of Binary Trees and Related Finite-Dimensional Algebras

The structure of a Hopf operad is defined on the vector spaces spanned by forests of leaf-labeled, rooted, binary trees. An explicit formula for the coproduct and its dual product is given, using a poset on forests.     

On Two Families of Hopf Algebras of Dimension 2 m

Abstract

In this paper we construct two families of semisimple Hopf algebras of dimension 2 ⁿ⁺¹, n ≥ 3. They are all constructed as Radford's biproducts. For these examples and their duals we compute their grouplike elements, centers, character algebras and Grothendieck rings. Comparing these facts we are able to show that depending on the dimension, representatives of one of the families are selfdual. We also prove that Hopf algebras from these families are neither triangular nor cotriangular and that their cocycle deformations are trivial.     

Diagrams of Hopf algebras with the Chevalley property

In this paper, we study non-cosemisimple Hopf algebras through their underlying coalgebra structure. We introduce the concept of the maximal pointed subcoalgebra/Hopf subalgebra. For a non-cosemisimple Hopf algebra A with the Chevalley property, if its diagram is a Nichols algebra, then the diagram of its maximal pointed Hopf subalgebra is also a Nichols algebra. When A is of finite dimension, we provide a necessary and sufficient condition for A’s diagram equaling the diagram of its maximal pointed Hopf subalgebra.     

Finite-Dimensional Algebras and Standard Systems

We introduce the notion of a standard system in order to deal with quasi-hereditary algebras. We shall prove that a necessary and sufficient condition for a finite-dimensional algebra to be quasi-hereditary is the existence of a full and divisible standard system. As a further application, we obtain a sufficient condition for a standardly stratified algebra.     

Finite-Dimensional Representations of Twisted Hyper-Loop Algebras

We investigate the category of finite-dimensional representations of twisted hyper-loop algebras, i.e., the hyperalgebras associated to twisted loop algebras over finite-dimensional simple Lie algebras. The main results are the classification of the irreducible modules, the definition of the universal highest-weight modules, called the Weyl modules, and, under a certain mild restriction on the characteristic of the ground field, a proof that the simple modules and the Weyl modules for the twisted hyper-loop algebras are isomorphic to appropriate simple and Weyl modules for the nontwisted hyper-loop algebras, respectively, via restriction of the action.     

Almost-triangular Hopf Algebras

In this paper, we consider a finite dimensional semisimple cosemisimple quasitriangular Hopf algebra with (we call this type of Hopf algebras almost-quasitriangular) over an algebraically closed field . We denote by the vector space generated by the left tensorand of . Then is a sub-Hopf algebra of . We proved that when is odd, has a triangular structure and can be obtained from a group algebra by twisting its usual comultiplication [14]; when is even, is an extension of an abelian group algebra and a triangular Hopf algebra, and may not be triangular. In general, an almost-triangular Hopf algebra can be viewed as a cocycle bicrossproduct.      

Hom-弱Hopf代数

在Hom-Hopf代数上,引入Hom-弱Hopf代数的概念并讨论了它的性质     

On the Codimension Growth of Finite-Dimensional Lie Algebras

We study the exponential growth of the codimensions c_n(L) of a finite-dimensional Lie algebra L over a field of characteristic zero. We show that if the solvable radical of L is nilpotent then exists and is an integer.     

Gradings on Finite-Dimensional Simple Lie Algebras

In this survey paper we present recent classification results for gradings by arbitrary groups on finite-dimensional simple Lie algebras over an algebraically closed field of characteristic different from 2. We also describe the main tools that were used to obtain these results (in particular, the classification of group gradings on matrix algebras).     

Prime-Dimensional Hopf Algebras

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On the antipode of Hopf algebras with the dual Chevalley property

In this paper, we study the antipode of a finite-dimensional Hopf algebra H with the dual Chevalley property and obtain an annihilation polynomial for the antipode. This generalizes an old result given by Taft and Wilson in 1974. As consequences, we show that 1) the quasi-exponent of H is the same as the exponent of its coradical, that is, $\mathrm{qexp}(H)=\exp ?(H_0)$; 2) $\mathrm{qexp}(H?\mathbb{k}〈S^2〉)=\mathrm{qexp}(H)$.     

Involutory Hopf Algebras with Non-Zero Integrals

The authors prove in this paper that an involutory Hopf algebrawith non-zero integral over a field of characteristic zero iscosemisimple. 2000 Mathematics Subject Classification 16W30.     

Gorenstein代数上的Hopf代数作用

令H是有限维Hopf代数，A是左H-模代数。本文证明了A是Gorenstein代数的充分必要条件。A^H也是Gorenstein代数的条件。它是Enochs EE,GarciaJJ和del RioA关于群作用相应的理论的推广，同时给出A/A^H是Frobenius扩张的条件。     

De-Equivariantization of Hopf Algebras

We study the de-equivariantization of a Hopf algebra by an affine group scheme and we apply Tannakian techniques in order to realize it as the tensor category of comodules over a coquasi-bialgebra. As an application we construct a family of coquasi-Hopf algebras A(H, G, Φ) attached to a coradically-graded pointed Hopf algebra H and some extra data.     

Pointed Hopf Algebras with Triangular Decomposition

In this paper, we present an approach to the definition of multiparameter quantum groups by studying Hopf algebras with triangular decomposition. Classifying all of these Hopf algebras which are of what we call weakly separable type over a group, we obtain a class of pointed Hopf algebras which can be viewed as natural generalizations of multiparameter deformations of universal enveloping algebras of Lie algebras. These Hopf algebras are instances of a new version of braided Drinfeld doubles, which we call asymmetric braided Drinfeld doubles. This is a generalization of an earlier result by Benkart and Witherspoon (Algebr. Represent. Theory 7(3) ? BC) who showed that two-parameter quantum groups are Drinfeld doubles. It is possible to recover a Lie algebra from these doubles in the case where the group is free abelian and the parameters are generic. The Lie algebras arising are generated by Lie subalgebras isomorphic to \(\mathfrak {sl}_{2}\).