scholarly journals GAUGE DEFORMATIONS FOR HOPF ALGEBRAS WITH THE DUAL CHEVALLEY PROPERTY

2012 ◽  
Vol 11 (03) ◽  
pp. 1250051
Author(s):  
ALESSANDRO ARDIZZONI ◽  
MARGARET BEATTIE ◽  
CLAUDIA MENINI
Keyword(s):  
Hopf Algebra ◽  
Gauge Transformation ◽  
Hopf Algebras ◽  
Characteristic Zero ◽  
Finite Dimensional ◽  
Chevalley Property ◽  
Algebra Over A Field

Let A be a Hopf algebra over a field K of characteristic zero such that its coradical H is a finite-dimensional sub-Hopf algebra. Our main theorem shows that there is a gauge transformation ζ on A such that Aζ ≅ Q#H where Aζ is the dual quasi-bialgebra obtained from A by twisting its multiplication by ζ, Q is a connected dual quasi-bialgebra in [Formula: see text] and Q#H is a dual quasi-bialgebra called the bosonization of Q by H.

HOPF ALGEBRAS OF DIMENSION 30

2010 ◽  
Vol 09 (01) ◽  
pp. 11-15 ◽  
Author(s):  
DAIJIRO FUKUDA
Keyword(s):  
Hopf Algebra ◽  
Group Algebra ◽  
Hopf Algebras ◽  
Characteristic Zero ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Finite Dimensional

This paper contributes to the classification of finite dimensional Hopf algebras. It is shown that every Hopf algebra of dimension 30 over an algebraically closed field of characteristic zero is semisimple and thus isomorphic to a group algebra or the dual of a group 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 Varieties of Null-Filiform Leibniz Algebras Under the Action of Hopf Algebras

2021 ◽  
Author(s):  
Lucio Centrone ◽  
Chia Zargeh
Keyword(s):  
Hopf Algebra ◽  
Free Algebra ◽  
Hopf Algebras ◽  
Leibniz Algebra ◽  
Module Structure ◽  
Leibniz Algebras ◽  
Finite Dimensional ◽  
Filiform Leibniz Algebra ◽  
Algebra Over A Field ◽  
Cocommutative Hopf Algebra

AbstractLet L be an n-dimensional null-filiform Leibniz algebra over a field K. We consider a finite dimensional cocommutative Hopf algebra or a Taft algebra H and we describe the H-actions on L. Moreover we provide the set of H-identities and the description of the Sn-module structure of the relatively free algebra of L.

scholarly journals Projectivity of Hopf algebras over subalgebras with semilocal central localizations

2008 ◽  
Vol 2 (1) ◽  
pp. 1-40 ◽  
Author(s):  
Serge Skryabin
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Module Algebra ◽  
Finitely Generated ◽  
Residually Finite ◽  
Hopf Subalgebra ◽  
Finite Dimensional ◽  
Algebra Over A Field

AbstractThe purpose of this paper is to extend the class of pairs A, H where H is a Hopf algebra over a field and A a right coideal subalgebra for which H is proved to be either projective or flat as an A-module. The projectivity is obtained under the assumption that H is residually finite dimensional, A has semilocal localizations with respect to a central subring, and there exists a Hopf subalgebra B of H such that the antipode of B is bijective and B is a finitely generated A-module. The flatness of H over A is shown to hold when H is a directed union of residually finite dimensional Hopf subalgebras, and there exists a Hopf subalgebra of H whose center contains A. More general projectivity and flatness results are established for (co)equivariant modules over an H-(co)module algebra under similar assumptions.

scholarly journals On the Semiprimitivity and Semiprimality Problems for Partial Smash Products

2018 ◽  
Vol 25 (01) ◽  
pp. 1-30
Author(s):  
Rafael Cavalheiro ◽  
Alveri Sant’Ana
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Irreducible Representations ◽  
Module Algebra ◽  
Perfect Field ◽  
Locally Finite ◽  
Semisimple Hopf Algebra ◽  
Finite Dimensional ◽  
Algebra Over A Field

In this paper we discuss about the semiprimitivity and the semiprimality of partial smash products. Let H be a semisimple Hopf algebra over a field 𝕜 and let A be a left partial H-module algebra. We study the H-prime and the H-Jacobson radicals of A and their relations with the prime and the Jacobson radicals of A#H, respectively. In particular, we prove that if A is H-semiprimitive, then A#H is semiprimitive provided that all irreducible representations of A are finite-dimensional, or A is an affine PI-algebra over 𝕜 and 𝕜 is a perfect field, or A is locally finite. Moreover, we prove that A#H is semiprime provided that A is an H-semiprime PI-algebra, generalizing to the setting of partial actions the known results for global actions of Hopf algebras.

scholarly journals A CLASS OF QUASITRIANGULAR GROUP-COGRADED MULTIPLIER HOPF ALGEBRAS

2018 ◽  
Vol 62 (1) ◽  
pp. 43-57
Author(s):  
TAO YANG ◽  
XUAN ZHOU ◽  
HAIXING ZHU
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Finite Dimensional ◽  
Special Cases ◽  
Multiplier Hopf Algebras ◽  
Multiplier Hopf Algebra ◽  
Classical Construction

AbstractFor a multiplier Hopf algebra pairing 〈A,B〉, we construct a class of group-cograded multiplier Hopf algebras D(A,B), generalizing the classical construction of finite dimensional Hopf algebras introduced by Panaite and Staic Mihai [Isr. J. Math. 158 (2007), 349–365]. Furthermore, if the multiplier Hopf algebra pairing admits a canonical multiplier in M(B⊗A) we show the existence of quasitriangular structure on D(A,B). As an application, some special cases and examples are provided.

scholarly journals The Drinfel’d double versus the Heisenberg double for Hom-Hopf algebras

2016 ◽  
Vol 15 (04) ◽  
pp. 1650059 ◽  
Author(s):  
Daowei Lu ◽  
Shuanhong Wang
Keyword(s):  
Group Theory ◽  
Hopf Algebra ◽  
Quantum Group ◽  
Hopf Algebras ◽  
Duke Math ◽  
Dual Pairs ◽  
Cambridge University ◽  
Drinfel’D Double ◽  
Finite Dimensional ◽  
Heisenberg Double

Let ([Formula: see text], [Formula: see text]) be a finite-dimensional Hom-Hopf algebra. In this paper we mainly construct the Drinfel’d double [Formula: see text] in the setting of Hom-Hopf algebras by two ways, one of which generalizes Majid’s bicrossproduct for Hopf algebras (see [S. Majid, Foundations of Quantum Group Theory (Cambridge University Press, 1995)]) and another one is to introduce the notion of dual pairs of Hom-Hopf algebras. Then we study the relation between the Drinfel’d double [Formula: see text] and Heisenberg double [Formula: see text], generalizing the main result in [J. H. Lu, On the Drinfel’d double and the Heisenberg double of a Hopf algebra, Duke Math. J. 74 (1994) 763–776]. The examples given in the paper are especially, not obtained from the usual Hopf algebras.

On coideal subalgebras of abelian cocentral extensions and a generalization of Wall's conjecture

2014 ◽  
Vol 14 (02) ◽  
pp. 1550021
Author(s):  
Sebastian Burciu
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Abelian Extension ◽  
Cyclic Module ◽  
Finite Dimensional ◽  
Dual Hopf Algebra ◽  
Finite Dimensional Hopf Algebra

It is shown that any coideal subalgebra of a finite-dimensional Hopf algebra is a cyclic module over the dual Hopf algebra. Using this we describe all coideal subalgebras of a cocentral abelian extension of Hopf algebras extending some results from [R. Guralnick and F. Xu, On a subfactor generalization of Wall's conjecture, J. Algebra 332 (2011) 457–468].

Ideals of Finite-Dimensional Pointed Hopf Algebras of Rank One

2021 ◽  
Vol 28 (02) ◽  
pp. 351-360
Author(s):  
Yu Wang ◽  
Zhihua Wang ◽  
Libin Li
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Adjoint Action ◽  
Prime Ideals ◽  
Closed Field ◽  
Finite Dimensional ◽  
Maximal Ideals ◽  
Primitive Ideals ◽  
Rank One ◽  
Pointed Hopf Algebras

Let [Formula: see text] 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 [Formula: see text]-module is generated by one element. In particular, any indecomposable submodule of [Formula: see text] under the adjoint action is generated by a special element of [Formula: see text]. Using this result, we show that the Hopf algebra [Formula: see text] is a principal ideal ring, i.e., any two-sided ideal of [Formula: see text] 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.

scholarly journals INVOLUTORY HOPF ALGEBRAS AND 3-MANIFOLD INVARIANTS

1991 ◽  
Vol 02 (01) ◽  
pp. 41-66 ◽  
Author(s):  
GREG KUPERBERG
Keyword(s):  
Hopf Algebra ◽  
Fundamental Group ◽  
Group Algebra ◽  
Hopf Algebras ◽  
Tensor Products ◽  
State Model ◽  
Formal Expression ◽  
Finite Dimensional ◽  
Heegaard Diagram ◽  
Structure Tensors

We establish a 3-manifold invariant for each finite-dimensional, involutory Hopf algebra. If the Hopf algebra is a group algebra G, the invariant counts homomorphisms from the fundamental group of the manifold to G. The invariant can be viewed as a state model on a Heegaard diagram or a triangulation of the manifold. The computation of the invariant involves tensor products and contractions of the structure tensors of the algebra. We show that every formal expression involving these tensors corresponds to a unique 3-manifold modulo a well-understood equivalence. This raises the possibility of an algorithm which can determine whether two given 3-manifolds are homeomorphic.

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