scholarly journals CLASSIFICATION OF PM QUIVER HOPF ALGEBRAS

2007 ◽  
Vol 06 (06) ◽  
pp. 919-950 ◽  
Author(s):  
SHOUCHUAN ZHANG ◽  
YAO-ZHONG ZHANG ◽  
HUI-XIANG CHEN
Keyword(s):  
Abelian Group ◽  
Hopf Algebras ◽  
Finite Abelian Group ◽  
Complex Field ◽  
Ground Field ◽  
Semisimple Lie Algebra ◽  
Enveloping Algebra ◽  
Complex Semisimple ◽  
Nichols Algebras

We describe certain quiver Hopf algebras by parameters. This leads to the classification of multiple Taft algebras as well as pointed Yetter–Drinfeld modules and their corresponding Nichols algebras. In particular, when the ground-field k is the complex field and G is a finite abelian group, we classify quiver Hopf algebras over G, multiple Taft algebras over G and Nichols algebras in [Formula: see text]. We show that the quantum enveloping algebra of a complex semisimple Lie algebra is a quotient of a semi-path Hopf algebra.

scholarly journals A connection between the number of subgroups and the order of a finite group

2020 ◽  
pp. 2250001
Author(s):  
Mihai-Silviu Lazorec
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Finite Abelian Group ◽  
Subgroup Lattice ◽  
Minimum Value ◽  
A Finite Group

For a finite group [Formula: see text], we associate the quantity [Formula: see text], where [Formula: see text] is the subgroup lattice of [Formula: see text]. Different properties and problems related to this ratio are studied throughout this paper. We determine the second minimum value of [Formula: see text] on the class of [Formula: see text]-groups of order [Formula: see text], where [Formula: see text] is an integer. We show that the set containing the quantities [Formula: see text], where [Formula: see text] is a finite (abelian) group, is dense in [Formula: see text] Finally, we consider [Formula: see text] to be a function on [Formula: see text] and we indicate some of its properties, the main result being the classification of finite abelian [Formula: see text]-groups [Formula: see text] satisfying [Formula: see text]

scholarly journals ON CYCLES AND COVERINGS ASSOCIATED TO A KNOT

2013 ◽  
Vol 22 (13) ◽  
pp. 1350074
Author(s):  
LILYA LYUBICH ◽  
MIKHAIL LYUBICH
Keyword(s):  
Dynamical System ◽  
Abelian Group ◽  
Finite Type ◽  
Weak Topology ◽  
Knot Theory ◽  
Commutator Subgroup ◽  
Finite Abelian Group ◽  
Complete Classification ◽  
And Shifts

Let [Formula: see text] be a knot, G be the knot group, K be its commutator subgroup, and x be a distinguished meridian. Let Σ be a finite abelian group. The dynamical system introduced by Silver and Williams in [Augmented group systems and n-knots, Math. Ann.296 (1993) 585–593; Augmented group systems and shifts of finite type, Israel J. Math.95 (1996) 231–251] consisting of the set Hom (K, Σ) of all representations ρ : K → Σ endowed with the weak topology, together with the homeomorphism [Formula: see text] is finite, i.e. it consists of several cycles. In [Periodic orbits of a dynamical system related to a knot, J. Knot Theory Ramifications20(3) (2011) 411–426] we found the lengths of these cycles for Σ = ℤ/p,p is prime, in terms of the roots of the Alexander polynomial of the knot, mod p. In this paper we generalize this result to a general abelian group Σ. This gives a complete classification of depth 2 solvable coverings over [Formula: see text].

scholarly journals Finite presentability of some metabelian Hopf algebras

2005 ◽  
Vol 72 (1) ◽  
pp. 109-127 ◽  
Author(s):  
Dessislava H. Kochloukova
Keyword(s):  
Abelian Group ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Hopf Algebras ◽  
Abelian Ideal ◽  
Finite Presentability ◽  
Finitely Presented ◽  
Homological Type

We classify the Hopf algebras U (L)#kQ of homological type FP2 where L is a Lie algebra and Q an Abelian group such that L has an Abelian ideal A invariant under the Q-action via conjugation and U (L/A)#kQ is commutative. This generalises the classification of finitely presented metabelian Lie algebras given by J. Groves and R. Bryant.

scholarly journals Groups containing locally maximal product-free sets of size 4

2021 ◽  
Vol 31 (2) ◽  
pp. 167-194
Author(s):  
C. S. Anabanti ◽  
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Finite Groups ◽  
Open Problem ◽  
Finite Abelian Group ◽  
Locally Maximal ◽  
A Finite Group ◽  
Free Set ◽  
Free Sets

Every locally maximal product-free set S in a finite group G satisfies G=S∪SS∪S−1S∪SS−1∪S−−√, where SS={xy∣x,y∈S}, S−1S={x−1y∣x,y∈S}, SS−1={xy−1∣x,y∈S} and S−−√={x∈G∣x2∈S}. To better understand locally maximal product-free sets, Bertram asked whether every locally maximal product-free set S in a finite abelian group satisfy |S−−√|≤2|S|. This question was recently answered in the negation by the current author. Here, we improve some results on the structures and sizes of finite groups in terms of their locally maximal product-free sets. A consequence of our results is the classification of abelian groups that contain locally maximal product-free sets of size 4, continuing the work of Street, Whitehead, Giudici and Hart on the classification of groups containing locally maximal product-free sets of small sizes. We also obtain partial results on arbitrary groups containing locally maximal product-free sets of size 4, and conclude with a conjecture on the size 4 problem as well as an open problem on the general case.

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 Separation of Variables for

2005 ◽  
Vol 48 (4) ◽  
pp. 587-600 ◽  
Author(s):  
Samuel A. Lopes
Keyword(s):  
Representation Theory ◽  
Lie Algebra ◽  
Analogous Result ◽  
Separation Of Variables ◽  
Positive Part ◽  
Semisimple Lie Algebra ◽  
Infinite Dimensional ◽  
Enveloping Algebra ◽  
Complex Semisimple ◽  
Quantized Enveloping Algebra

AbstractLet be the positive part of the quantized enveloping algebra . Using results of Alev–Dumas and Caldero related to the center of , we show that this algebra is free over its center. This is reminiscent of Kostant's separation of variables for the enveloping algebra U(g) of a complex semisimple Lie algebra g, and also of an analogous result of Joseph–Letzter for the quantum algebra Ŭq(g). Of greater importance to its representation theory is the fact that is free over a larger polynomial subalgebra N in n variables. Induction from N to provides infinite-dimensional modules with good properties, including a grading that is inherited by submodules.

scholarly journals KRULL DIMENSION OF AFFINOID ENVELOPING ALGEBRAS OF SEMISIMPLE LIE ALGEBRAS

2013 ◽  
Vol 55 (A) ◽  
pp. 7-26
Author(s):  
KONSTANTIN ARDAKOV ◽  
IAN GROJNOWSKI
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Krull Dimension ◽  
Semisimple Lie Algebra ◽  
Semisimple Lie Algebras ◽  
Enveloping Algebras ◽  
Enveloping Algebra ◽  
Complex Semisimple

AbstractUsing Beilinson–Bernstein localisation, we give another proof of Levasseur's theorem on the Krull dimension of the enveloping algebra of a complex semisimple Lie algebra. The proof also extends to the case of affinoid enveloping algebras.

scholarly journals On finite GK-dimensional Nichols algebras over abelian groups

2021 ◽  
Vol 271 (1329) ◽  
Author(s):  
Nicolás Andruskiewitsch ◽  
Iván Angiono ◽  
István Heckenberger
Keyword(s):  
Hopf Algebras ◽  
Abelian Groups ◽  
Jordan Block ◽  
Vector Spaces ◽  
Direct Sums ◽  
Content Type ◽  
Nichols Algebra ◽  
Nichols Algebras ◽  
Kirillov Dimension

We contribute to the classification of Hopf algebras with finite Gelfand-Kirillov dimension, GKdim \operatorname {GKdim} for short, through the study of Nichols algebras over abelian groups. We deal first with braided vector spaces over Z \mathbb {Z} with the generator acting as a single Jordan block and show that the corresponding Nichols algebra has finite GKdim \operatorname {GKdim} if and only if the size of the block is 2 and the eigenvalue is ± 1 \pm 1 ; when this is 1, we recover the quantum Jordan plane. We consider next a class of braided vector spaces that are direct sums of blocks and points that contains those of diagonal type. We conjecture that a Nichols algebra of diagonal type has finite GKdim \operatorname {GKdim} if and only if the corresponding generalized root system is finite. Assuming the validity of this conjecture, we classify all braided vector spaces in the mentioned class whose Nichols algebra has finite GKdim \operatorname {GKdim} . Consequently we present several new examples of Nichols algebras with finite GKdim \operatorname {GKdim} , including two not in the class alluded to above. We determine which among these Nichols algebras are domains.

scholarly journals CLASSIFICATION OF QUIVER HOPF ALGEBRAS AND POINTED HOPF ALGEBRAS OF TYPE ONE

2012 ◽  
Vol 87 (2) ◽  
pp. 216-237
Author(s):  
SHOUCHUAN ZHANG ◽  
HUI-XIANG CHEN ◽  
YAO-ZHONG ZHANG
Keyword(s):  
Hopf Algebras ◽  
Irreducible Representations ◽  
Group Algebras ◽  
Pointed Hopf Algebras ◽  
Nichols Algebras

AbstractQuiver Hopf algebras are classified by means of ramification systems with irreducible representations. This leads to the classification of Nichols algebras over group algebras and pointed Hopf algebras of type one.

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