The corepresentation ring of a pointed Hopf algebra of rank one

2018 ◽  
Vol 17 (12) ◽  
pp. 1850236
Author(s):  
Zhihua Wang
Keyword(s):  
Abelian Group ◽  
Hopf Algebra ◽  
Nilpotent Element ◽  
Finite Abelian Group ◽  
Generators And Relations ◽  
Direct Sum ◽  
Ring Form ◽  
Nilpotent Elements ◽  
Finite Dimensional ◽  
Rank One

Let [Formula: see text] be an arbitrary pointed Hopf algebra of rank one and [Formula: see text] the group of group-like elements of [Formula: see text]. In this paper, we give the decomposition of a tensor product of finite dimensional indecomposable right [Formula: see text]-comodules into a direct sum of indecomposables. This enables us to describe the corepresentation ring of [Formula: see text] in terms of generators and relations. Such a ring is not commutative if [Formula: see text] is not abelian. We describe all nilpotent elements of the corepresentation ring of [Formula: see text] if [Formula: see text] is a finite abelian group or a particular Hamiltonian group. In this case, all nilpotent elements of the corepresentation ring form a principal ideal which is either zero or generated by a nilpotent element of degree 2.

scholarly journals On the group ring of a finite abelian group

1969 ◽  
Vol 1 (2) ◽  
pp. 245-261 ◽  
Author(s):  
Raymond G. Ayoub ◽  
Christine Ayoub
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Group Ring ◽  
Free Abelian Group ◽  
Finite Abelian Group ◽  
Rational Numbers ◽  
Cyclotomic Fields ◽  
Group Of Units ◽  
Direct Sum ◽  
A Finite Group

The group ring of a finite abelian group G over the field of rational numbers Q and over the rational integers Z is studied. A new proof of the fact that the group ring QG is a direct sum of cyclotomic fields is given – without use of the Maschke and Wedderburn theorems; it is shown that the projections of QG onto these fields are determined by the inequivalent characters of G. It is proved that the group of units of ZG is a direct product of a finite group and a free abelian group F and the rank of F is determined. A formula for the orthogonal idempotents of QG is found.

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 Identities and central polynomials for real graded division algebras

2017 ◽  
Vol 27 (07) ◽  
pp. 935-952
Author(s):  
Diogo Diniz ◽  
Claudemir Fidelis ◽  
Sérgio Mota
Keyword(s):  
Abelian Group ◽  
Finite Abelian Group ◽  
Finite Basis ◽  
Division Algebras ◽  
Polynomial Identities ◽  
Finite Dimensional ◽  
Real Algebra ◽  
Graded Polynomial Identities

Let [Formula: see text] be a finite dimensional simple real algebra with a division grading by a finite abelian group [Formula: see text]. In this paper, we provide a finite basis for the [Formula: see text]-ideal of graded polynomial identities for [Formula: see text] and a finite basis for the [Formula: see text]-space of graded central polynomials for [Formula: see text].

scholarly journals Classification of Finite Rings of Orderp6by Generators and Relations

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
P. Karimi Beiranvand ◽  
R. Beyranvand ◽  
M. Gholami
Keyword(s):  
Abelian Group ◽  
Prime Number ◽  
Binary Operation ◽  
Finite Abelian Group ◽  
Sufficient Conditions ◽  
Additive Group ◽  
Finite Rings ◽  
Generators And Relations ◽  
Necessary And Sufficient

For any finite abelian group(R,+), we define a binary operation or “multiplication” onRand give necessary and sufficient conditions on this multiplication forRto extend to a ring. Then we show when two rings made on the same group are isomorphic. In particular, it is shown that there aren+1rings of orderpnwith characteristicpn, wherepis a prime number. Also, all finite rings of orderp6are described by generators and relations. Finally, we give an algorithm for the computation of all finite rings based on their additive group.

scholarly journals INTERCHANGE RINGS

2016 ◽  
Vol 101 (3) ◽  
pp. 310-334
Author(s):  
CHARLES C. EDMUNDS
Keyword(s):  
Abelian Group ◽  
Prime Power ◽  
Binary Operation ◽  
Finite Abelian Group ◽  
Linear Operators ◽  
Prime Power Order ◽  
Cyclic Groups ◽  
Isomorphism Classes ◽  
Direct Sum ◽  
Interchange Law

An interchange ring,$(R,+,\bullet )$, is an abelian group with a second binary operation defined so that the interchange law$(w+x)\bullet (y+z)=(w\bullet y)+(x\bullet z)$ holds. An interchange near ring is the same structure based on a group which may not be abelian. It is shown that each interchange (near) ring based on a group $G$ is formed from a pair of endomorphisms of $G$ whose images commute, and that all interchange (near) rings based on $G$ can be characterized in this manner. To obtain an associative interchange ring, the endomorphisms must be commuting idempotents in the endomorphism semigroup of $G$. For $G$ a finite abelian group, we develop a group-theoretic analogue of the simultaneous diagonalization of idempotent linear operators and show that pairs of endomorphisms which yield associative interchange rings can be diagonalized and then put into a canonical form. A best possible upper bound of $4^{r}$ can be given for the number of distinct isomorphism classes of associative interchange rings based on a finite abelian group $A$ which is a direct sum of $r$ cyclic groups of prime power order. If $A$ is a direct sum of $r$ copies of the same cyclic group of prime power order, we show that there are exactly ${\textstyle \frac{1}{6}}(r+1)(r+2)(r+3)$ distinct isomorphism classes of associative interchange rings based on $A$. Several examples are given and further comments are made about the general theory of interchange rings.

scholarly journals Quantitative norm convergence of double ergodic averages associated with two commuting group actions

2014 ◽  
Vol 36 (3) ◽  
pp. 860-874 ◽  
Author(s):  
VJEKOSLAV KOVAČ
Keyword(s):  
Abelian Group ◽  
Finite Abelian Group ◽  
Group Actions ◽  
Norm Convergence ◽  
Ergodic Averages ◽  
Direct Sum ◽  
Measure Preserving Actions

We study double averages along orbits for measure-preserving actions of$\mathbb{A}^{{\it\omega}}$, the direct sum of countably many copies of a finite abelian group$\mathbb{A}$. We show an$\text{L}^{p}$norm-variation estimate for these averages, which in particular re-proves their convergence in$\text{L}^{p}$for any finite$p$and for any choice of two$\text{L}^{\infty }$functions. The result is motivated by recent questions on quantifying convergence of multiple ergodic averages.

On Finite Essential Extensions of Torsion Free Abelian Groups

1996 ◽  
Vol 48 (5) ◽  
pp. 918-929 ◽  
Author(s):  
K. Benabdallah ◽  
M. A. Ouldbeddi
Keyword(s):  
Abelian Group ◽  
Finite Index ◽  
Free Abelian Group ◽  
Torsion Free Abelian Group ◽  
Direct Sum ◽  
Rank One ◽  
Free Abelian Groups ◽  
Decomposable Groups ◽  
Essential Extensions ◽  
Torsion Free

AbstractLet A be a torsion free abelian group. We say that a group K is a finite essential extension of A if K contains an essential subgroup of finite index which is isomorphic to A. Such K admits a representation as (A ℤ xkx)/ℤky where y = Nx + a for some k x k matrix N over Z and α ∈ Ak satisfying certain conditions of relative primeness and ℤk = {(α1,..., αk) : αi, ∈ ℤ}. The concept of absolute width of an f.e.e. K of A is defined and it is shown to be strictly smaller than the rank of A. A kind of basis substitution with respect to Smith diagonal matrices is shown to hold for homogeneous completely decomposable groups. This result together with general properties of our representations are used to provide a self contained proof that acd groups with two critical types are direct sum of groups of rank one and two.

scholarly journals Handlebody-knot invariants derived from unimodular Hopf algebras

2014 ◽  
Vol 23 (07) ◽  
pp. 1460001 ◽  
Author(s):  
Atsushi Ishii ◽  
Akira Masuoka
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Tensor Category ◽  
Symmetric Algebra ◽  
Computational Results ◽  
Drinfeld Modules ◽  
Generators And Relations ◽  
Knot Invariants ◽  
Finite Dimensional ◽  
Tensor Functor

To systematically construct invariants of handlebody-links, we give a new presentation of the braided tensor category [Formula: see text] of handlebody-tangles by generators and relations, and prove that given what we call a quantum-commutative quantum-symmetric algebra A in an arbitrary braided tensor category [Formula: see text], there arises a braided tensor functor [Formula: see text], which gives rise to a desired invariant. Some properties of the invariants and explicit computational results are shown especially when A is a finite-dimensional unimodular Hopf algebra, which is naturally regarded as a quantum-commutative quantum-symmetric algebra in the braided tensor category [Formula: see text] of Yetter–Drinfeld modules.

scholarly journals Inverse results for weighted Harborth constants

2016 ◽  
Vol 12 (07) ◽  
pp. 1845-1861 ◽  
Author(s):  
Luz E. Marchan ◽  
Oscar Ordaz ◽  
Dennys Ramos ◽  
Wolfgang A. Schmid
Keyword(s):  
Inverse Problem ◽  
Abelian Group ◽  
Inverse Problems ◽  
Cyclic Group ◽  
Finite Abelian Group ◽  
Maximal Length ◽  
Direct Sum ◽  
Inverse Results

For a finite abelian group [Formula: see text], the Harborth constant is defined as the smallest integer [Formula: see text] such that each squarefree sequence over [Formula: see text] of length [Formula: see text] has a subsequence of length equal to the exponent of [Formula: see text] whose terms sum to [Formula: see text]. The plus-minus weighted Harborth constant is defined in the same way except that the existence of a plus-minus weighted subsum equaling [Formula: see text] is required, that is, when forming the sum one can choose a sign for each term. The inverse problem associated to these constants is the problem of determining the structure of squarefree sequences of maximal length that do not yet have such a zero-subsum. We solve the inverse problems associated to these constants for certain groups, in particular, for groups that are the direct sum of a cyclic group and a group of order two. Moreover, we obtain some results for the plus-minus weighted Erdős–Ginzburg–Ziv constant.

scholarly journals On decomposable pseudofree groups

1999 ◽  
Vol 22 (3) ◽  
pp. 617-628
Author(s):  
Dirk Scevenels
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
The Other ◽  
Direct Sum ◽  
Other Hand ◽  
Rank One ◽  
Free Abelian Groups

An Abelian group is pseudofree of rankℓif it belongs to the extended genus ofℤℓ, i.e., its localization at every primepis isomorphic toℤpℓ. A pseudofree group can be studied through a sequence of rational matrices, the so-called sequential representation. Here, we use these sequential representations to study the relation between the product of extended genera of free Abelian groups and the extended genus of their direct sum. In particular, using sequential representations, we give a new proof of a result by Baer, stating that two direct sum decompositions into rank one groups of a completely decomposable pseudofree Abelian group are necessarily equivalent. On the other hand, sequential representations can also be used to exhibit examples of pseudofree groups having nonequivalent direct sum decompositions into indecomposable groups. However, since this cannot occur when using the notion of near-isomorphism rather than isomorphism, we conclude our work by giving a characterization of near-isomorphism for pseudofree groups in terms of their sequential representations.

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