ABSTRACT ALGEBRAIC INTEGRALS AND FROBENIUS CATEGORIES

2013 ◽  
Vol 24 (10) ◽  
pp. 1350081 ◽  
Author(s):  
MIODRAG CRISTIAN IOVANOV
Keyword(s):  
Invariant Measure ◽  
Hopf Algebra ◽  
Hopf Algebras ◽  
Existence And Uniqueness ◽  
Compact Groups ◽  
Finite Dimensional ◽  
Frobenius Categories ◽  
Finite Dimensional Algebras

We generalize results on the connection between existence and uniqueness of integrals and representation theoretic properties for Hopf algebras and compact groups. For this, given a coalgebra C, we study analogues of the existence and uniqueness properties for the integral functor Hom C(C, -), which generalizes the notion of integral in a Hopf algebra. We show that the coalgebra C is co-Frobenius if and only if dim ( Hom C(C, M)) = dim (M) for all finite dimensional right (left) comodules M. As applications, we give a few new categorical characterizations of co-Frobenius, quasi-co-Frobenius (QcF) coalgebras and semiperfect coalgebras, and re-derive classical results of Lin, Larson, Sweedler and Sullivan on Hopf algebras. We show that a coalgebra is QcF if and only if the category of left (right) comodules is Frobenius, generalizing results from finite dimensional algebras, and we show that a one-sided QcF coalgebra is two-sided semiperfect. We also construct a class of examples derived from quiver coalgebras to show that the results of the paper are the best possible. Finally, we examine the case of compact groups, and note that algebraic integrals can be interpreted as certain skew-invariant measure theoretic integrals on the group.

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.

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.

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].

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.

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.

WEYL ALGEBRAS, FOURIER TRANSFORMATIONS AND INTEGRALS ON FINITE-DIMENSIONAL HOPF ALGEBRAS

1994 ◽  
Vol 06 (01) ◽  
pp. 149-166 ◽  
Author(s):  
FLORIAN NILL
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Multiplication Operators ◽  
Hermitian Forms ◽  
Weyl Algebras ◽  
Finite Dimensional ◽  
Natural Choice ◽  
Translation Operators ◽  
Canonical Representations ◽  
Left And Right

The "Weyl algebras" [Formula: see text], σ, σ′ ∈ {+, –}, over a finite-dimensional Hopf algebra H are defined to be the abstract algebras generated by left or right (σ = ±) multiplication operators Qσ (ψ), ψ ∈ H, and left or right (σ′ = ±) translation operators Pσ′ (a), [Formula: see text] dual of H. It is shown that these algebras are all isomorphic to End (H). Fourier transformations are defined as intertwiners [Formula: see text] implementing a natural isomorphism [Formula: see text]. As a byproduct, this formalism provides a new proof of the invertibility of the antipode and the uniqueness (up to multiplication by a constant) of left and right integrals on finite-dimensional Hopf algebras. If H is a star Hopf algebra, the canonical representations of [Formula: see text] become star representation and Fourier transformation becomes an isometry w.r.t. a natural choice of nondegenerate hermitian forms on H and [Formula: see text]. Some comments on the relation with the C*-approach of Woronowicz and Podleś & Woronowicz are added.

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.

Sign in / Sign up

Export Citation Format

Share Document