scholarly journals The combinatorics of an algebraic class of easy quantum groups

2014 ◽  
Vol 17 (03) ◽  
pp. 1450016 ◽  
Author(s):  
Sven Raum ◽  
Moritz Weber
Keyword(s):  
Symmetric Group ◽  
Quantum Group ◽  
Permutation Group ◽  
Free Product ◽  
Quantum Groups ◽  
Orthogonal Group ◽  
Point Of View ◽  
Algebraic Point ◽  
Matrix Quantum

Easy quantum groups are compact matrix quantum groups, whose intertwiner spaces are given by the combinatorics of categories of partitions. This class contains the symmetric group Sn and the orthogonal group On as well as Wang's quantum permutation group [Formula: see text] and his free orthogonal quantum group [Formula: see text]. In this paper, we study a particular class of categories of partitions to each of which we assign a subgroup of the infinite free product of the cyclic group of order two. This is an important step in the classification of all easy quantum groups and we deduce that there are uncountably many of them. We focus on the combinatorial aspects of this assignment, complementing the quantum algebraic point of view presented in another paper.

A lattice of extension rings for a commutative ring

1978 ◽  
Vol 84 (2) ◽  
pp. 225-234 ◽  
Author(s):  
D. Kirby ◽  
M. R. Adranghi
Keyword(s):  
Point Of View ◽  
Local Rings ◽  
Algebraic Point ◽  
Multiple Points ◽  
One Dimensional ◽  
Maximal Ideals ◽  
Blowing Up ◽  
Abstract Case ◽  
Algebroid Curve

The work of this note was motivated in the first place by North-cott's theory of dilatations for one-dimensional local rings (see, for example (4) and (5)). This produces a tree of local rings as in (4) which corresponds, in the abstract case, to the branching sequence of infinitely-near multiple points on an algebroid curve. From the algebraic point of view it seems more natural to characterize such one-dimensional local rings R by means of the set of rings which arise by blowing up all ideals Q which are primary for the maximal ideals M of R. This set of rings forms a lattice (R), ordered by inclusion, each ring S of which is a finite R-module. Moreover the length of the R-module S/R is just the reduction number of the corresponding ideal Q (cf. theorem 1 of Northcott (6)). Thus the lattice (R) provides a finer classification of the rings R than does the set of reduction numbers (cf. Kirby (1)).

scholarly journals Classification of globally colorized categories of partitions

2018 ◽  
Vol 21 (04) ◽  
pp. 1850029 ◽  
Author(s):  
Daniel Gromada
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Tensor Category ◽  
Set Partitions

Set partitions closed under certain operations form a tensor category. They give rise to certain subgroups of the free orthogonal quantum group [Formula: see text], the so-called easy quantum groups, introduced by Banica and Speicher in 2009. This correspondence was generalized to two-colored set partitions, which, in addition, assign a black or white color to each point of a set. Globally colorized categories of partitions are those categories that are invariant with respect to arbitrary permutations of colors. This paper presents a classification of globally colorized categories. In addition, we show that the corresponding unitary quantum groups can be constructed from the orthogonal ones using tensor complexification.

scholarly journals PARAGRASSMANN ANALYSIS AND QUANTUM GROUPS

1992 ◽  
Vol 07 (23) ◽  
pp. 2129-2141 ◽  
Author(s):  
A. T. FILIPPOV ◽  
A. P. ISAEV ◽  
A. B. KURDIKOV
Keyword(s):  
Differential Operator ◽  
Quantum Groups ◽  
Natural Generalization ◽  
Point Of View ◽  
Algebraic Point ◽  
Root Of Unity ◽  
Deformation Parameters ◽  
One And Many

Paragrassmann algebras with one and many paragrassmann variables are considered from the algebraic point of view without using the Green ansatz. A differential operator with respect to paragrassmann variable and a covariant para-super-derivative are introduced giving a natural generalization of the Grassmann calculus to a paragrassmann one. Deep relations between paragrassmann algebras and quantum groups with deformation parameters being root of unity are established.

scholarly journals WHERE ARE ELKO SPINOR FIELDS IN LOUNESTO SPINOR FIELD CLASSIFICATION?

2006 ◽  
Vol 21 (01) ◽  
pp. 65-74 ◽  
Author(s):  
R. DA ROCHA ◽  
W. A. RODRIGUES
Keyword(s):  
Spinor Field ◽  
Point Of View ◽  
Algebraic Point ◽  
General Classification ◽  
Spinor Fields ◽  
Majorana Spinor ◽  
Algebraic Constraints ◽  
Elko Spinor Fields ◽  
Component Spinor

This paper proves that from the algebraic point of view ELKO spinor fields belong together with Majorana spinor fields to a wider class, the so-called flagpole spinor fields, corresponding to the class 5, according to Lounesto spinor field classification. We show moreover that algebraic constraints imply that any class 5 spinor field is such that the 2-component spinor fields entering its structure have opposite helicities. The proof of our statement is based on Lounesto general classification of all spinor fields, according to the relations and values taken by their associated bilinear covariants, and can eventually shed some new light on the algebraic investigations concerning dark matter.

ON MATRIX QUANTUM GROUPS OF TYPE An

2000 ◽  
Vol 11 (09) ◽  
pp. 1115-1146 ◽  
Author(s):  
HO Hai PHUNG
Keyword(s):  
Hopf Algebra ◽  
Quantum Group ◽  
Quantum Groups ◽  
Semi Group ◽  
The Matrix ◽  
Hecke Symmetry ◽  
Matrix Quantum

Given a Hecke symmetry R, one can define a matrix bialgebra ER and a matrix Hopf algebra HR, which are called function rings on the matrix quantum semi-group and matrix quantum groups associated to R. We show that for an even Hecke symmetry, the rational representations of the corresponding quantum group are absolutely reducible and that the fusion coefficients of simple representations depend only on the rank of the Hecke symmetry. Further we compute the quantum rank of simple representations. We also show that the quantum semi-group is "Zariski" dense in the quantum group. Finally we give a formula for the integral.

scholarly journals On the classification of the Lie algebrasLr,ts

2004 ◽  
Vol 2004 (42) ◽  
pp. 2269-2272
Author(s):  
L. A-M. Hanna
Keyword(s):  
Lie Algebras ◽  
Isomorphism Class ◽  
Matrix Representation ◽  
Point Of View ◽  
Algebraic Point

The Lie algebrasLr,tsintroduced by the author (2003) are classified from an algebraic point of view. A matrix representation of least degree is given for each isomorphism class.

scholarly journals Classification of Integral Modular Categories of Frobenius–Perron Dimension pq4 and p2q2

2014 ◽  
Vol 57 (4) ◽  
pp. 721-734 ◽  
Author(s):  
Paul Bruillard ◽  
Cásar Galindo ◽  
Seung-Moon Hong ◽  
Yevgenia Kashina ◽  
Deepak Naidu ◽  
...  
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Fusion Categories ◽  
Modular Category

AbstractWe classify integral modular categories of dimension pq4 and p2q2, where p and q are distinct primes. We show that such categories are always group-theoretical, except for categories of dimension 4q2. In these cases there are well-known examples of non-group-theoretical categories, coming from centers of Tambara–Yamagami categories and quantum groups. We show that a non-grouptheoretical integral modular category of dimension 4q2 is either equivalent to one of these well-known examples or is of dimension 36 and is twist-equivalent to fusion categories arising froma certain quantum group.

scholarly journals UNIVERSAL QUANTUM GROUPS

1996 ◽  
Vol 07 (02) ◽  
pp. 255-263 ◽  
Author(s):  
ALFONS VAN DAELE ◽  
SHUZHOU WANG
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Universal Quantum ◽  
Group A ◽  
Matrix Quantum

For each invertible m×m matrix Q a compact matrix quantum group Au(Q) is constructed. These quantum groups are shown to be universal in the sense that any compact matrix quantum group is a quantum subgroup of some of them. Their orthogonal version Ao(Q) is also constructed. Finally, we discuss related constructions in the literature.

REPRESENTATIONS OF QUANTUM GROUPS AT ROOTS OF 1: REDUCTION TO THE EXCEPTIONAL CASE

1992 ◽  
Vol 07 (supp01a) ◽  
pp. 141-149 ◽  
Author(s):  
CORRADO DE CONCINI ◽  
VICTOR G. KAC
Keyword(s):  
Irreducible Representation ◽  
Quantum Group ◽  
Quantum Groups ◽  
Exceptional Case

This paper is a continuation of the papers [DC-K] and [DC-K-P] on representations of quautum groups at roots of 1. Here we show that an irreducible representation of a quantum group at an odd root of 1 can be uniquely induced from an exceptional representation of a smaller quantum group. This reduces the classification of representations, the calculation of their characters and dimensions, etc, to the exceptional case.

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