scholarly journals ON REPRESENTATIONS OF QUANTUM GROUPS Uq(fm(K,H))

2008 ◽  
Vol 78 (2) ◽  
pp. 261-284 ◽  
Author(s):  
XIN TANG ◽  
YUNGE XU
Keyword(s):  
Quantum Groups ◽  
Irreducible Representations ◽  
Whittaker Model ◽  
Finite Dimensional ◽  
Completely Reducible ◽  
The Relationship ◽  
Natural Families

AbstractWe construct families of irreducible representations for a class of quantum groups Uq(fm(K,H). First, we realize these quantum groups as hyperbolic algebras. Such a realization yields natural families of irreducible weight representations for Uq(fm(K,H)). Second, we study the relationship between Uq(fm(K,H)) and Uq(fm(K)). As a result, any finite-dimensional weight representation of Uq(fm(K,H)) is proved to be completely reducible. Finally, we study the Whittaker model for the center of Uq(fm(K,H)), and a classification of all irreducible Whittaker representations of Uq(fm(K,H)) is obtained.

ON PERIODIC REPRESENTATIONS OF QUANTUM GROUPS

1992 ◽  
Vol 06 (11n12) ◽  
pp. 1873-1880
Author(s):  
DANIEL ARNAUDON
Keyword(s):  
Lie Algebra ◽  
Quantum Groups ◽  
Irreducible Representations ◽  
Root Of Unity ◽  
Dimension Formula ◽  
Finite Dimensional ◽  
Affine Lie Algebra

We present some results on representations of quantum groups at the root of unity. In the case of SL(2)q, the classification of the finite dimensional irreducible representations is given. For [Formula: see text] with [Formula: see text] a semi-simple or affine Lie algebra and q an mth root of unity (m odd), we classify the representations of dimension [Formula: see text] on which the actions of the Chevalley generators are injective. Finally, we adapt the Gelfand-Zetlin basis to the case of SL(N)q and IU(N)q.

scholarly journals On the Structure of Finite Groupoids and Their Representations

Symmetry ◽  
2019 ◽  
Vol 11 (3) ◽  
pp. 414 ◽  
Author(s):  
Alberto Ibort ◽  
Miguel Rodríguez
Keyword(s):  
Irreducible Representation ◽  
Regular Representation ◽  
Irreducible Representations ◽  
Quantum Mechanical ◽  
Linear Representations ◽  
Finite Dimensional ◽  
Completely Reducible ◽  
Quantum Mechanical Systems ◽  
Maschke’S Theorem ◽  
Totally Disconnected

In this paper, both the structure and the theory of representations of finite groupoids are discussed. A finite connected groupoid turns out to be an extension of the groupoids of pairs of its set of units by its canonical totally disconnected isotropy subgroupoid. An extension of Maschke’s theorem for groups is proved showing that the algebra of a finite groupoid is semisimple and all finite-dimensional linear representations of finite groupoids are completely reducible. The theory of characters for finite-dimensional representations of finite groupoids is developed and it is shown that irreducible representations of the groupoid are in one-to-one correspondence with irreducible representation of its isotropy groups, with an extension of Burnside’s theorem describing the decomposition of the regular representation of a finite groupoid. Some simple examples illustrating these results are exhibited with emphasis on the groupoids interpretation of Schwinger’s description of quantum mechanical systems.

Classification of irreducible representations of Lie algebra of vector fields on a torus

2016 ◽  
Vol 2016 (720) ◽  
Author(s):  
Yuly Billig ◽  
Vyacheslav Futorny
Keyword(s):  
Lie Algebra ◽  
Vector Fields ◽  
Irreducible Representations ◽  
Simple Modules ◽  
Finite Dimensional ◽  
Standing Problem

AbstractWe solve a long standing problem of the classification of all simple modules with finite-dimensional weight spaces over Lie algebra of vector fields on

scholarly journals Finite quasi-quantum groups of diagonal type

2020 ◽  
Vol 2020 (759) ◽  
pp. 201-243 ◽  
Author(s):  
Hua-Lin Huang ◽  
Gongxiang Liu ◽  
Yuping Yang ◽  
Yu Ye
Keyword(s):  
Quantum Groups ◽  
Hopf Algebras ◽  
Tensor Categories ◽  
Finite Dimensional

AbstractIn this paper, we give a classification of finite-dimensional radically graded elementary quasi-Hopf algebras of diagonal type, or equivalently, finite-dimensional coradically graded pointed Majid algebras of diagonal type. By a Tannaka–Krein type duality, this determines a big class of pointed finite tensor categories. Some efficient methods of construction are also given.

IRREDUCIBLE REPRESENTATIONS OF A CERTAIN JORDAN SUPERALGEBRA

2005 ◽  
Vol 04 (01) ◽  
pp. 1-14 ◽  
Author(s):  
M. N. TRUSHINA
Keyword(s):  
Irreducible Representations ◽  
Jordan Superalgebra ◽  
Finite Dimensional ◽  
Simple Superalgebra

A classification of unital finite-dimensional irreducible Jordan representations of the simple superalgebra D(t) in the case of characteristic 0 was obtained. As a corollary we obtain a classification of finite-dimensional irreducible representations of the Kaplansky superalgebra K3 in the case of characteristic 0.

scholarly journals Classification theorem on irreducible representations of theq-deformed algebraU′q(son)

2005 ◽  
Vol 2005 (2) ◽  
pp. 225-262 ◽  
Author(s):  
N. Z. Iorgov ◽  
A. U. Klimyk
Keyword(s):  
Lie Algebra ◽  
Complete Classification ◽  
Universal Enveloping Algebra ◽  
Irreducible Representations ◽  
Classical Type ◽  
Enveloping Algebra ◽  
Root Of Unity ◽  
Complete Reducibility ◽  
Finite Dimensional

The aim of this paper is to give a complete classification of irreducible finite-dimensional representations of the nonstandardq-deformationU′q(son)(which does not coincide with the Drinfel'd-Jimbo quantum algebraUq(son)) of the universal enveloping algebraU(son(ℂ))of the Lie algebrason(ℂ)whenqis not a root of unity. These representations are exhausted by irreducible representations of the classical type and of the nonclassical type. The theorem on complete reducibility of finite-dimensional representations ofU′q(son)is proved.

scholarly journals Towards a classification of finite-dimensional representations of rational Cherednik algebras of type D

2018 ◽  
Vol 17 (12) ◽  
pp. 1850237
Author(s):  
Seth Shelley-Abrahamson ◽  
Alec Sun
Keyword(s):  
Irreducible Representations ◽  
Infinite Dimensional ◽  
Cherednik Algebras ◽  
Type Formula ◽  
Type D ◽  
Finite Dimensional ◽  
Rational Cherednik Algebra ◽  
Wall Crossing ◽  
Rational Cherednik Algebras

Using a combinatorial description due to Jacon and Lecouvey of the wall crossing bijections for cyclotomic rational Cherednik algebras, we show that the irreducible representations [Formula: see text] of the rational Cherednik algebra [Formula: see text] of type [Formula: see text] for symmetric bipartitions [Formula: see text] are infinite dimensional for all parameters [Formula: see text]. In particular, all finite dimensional irreducible representations of rational Cherednik algebras of type [Formula: see text] arise as restrictions of finite-dimensional irreducible representations of rational Cherednik algebras of type [Formula: see text].

scholarly journals BIDIAGONAL PAIRS, THE LIE ALGEBRA 𝔰𝔩2, AND THE QUANTUM GROUP Uq(𝔰𝔩2)

2013 ◽  
Vol 12 (05) ◽  
pp. 1250207 ◽  
Author(s):  
DARREN FUNK-NEUBAUER
Keyword(s):  
Quantum Group ◽  
Lie Algebra ◽  
Dimensional Representation ◽  
Linear Transformations ◽  
Finite Dimensional Representation ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
The Relationship

We introduce a linear algebraic object called a bidiagonal pair. Roughly speaking, a bidiagonal pair is a pair of diagonalizable linear transformations on a finite-dimensional vector space, each of which acts in a bidiagonal fashion on the eigenspaces of the other. We associate to each bidiagonal pair a sequence of scalars called a parameter array. Using this concept of a parameter array we present a classification of bidiagonal pairs up to isomorphism. The statement of this classification does not explicitly mention the Lie algebra 𝔰𝔩2 or the quantum group Uq(𝔰𝔩2). However, its proof makes use of the finite-dimensional representation theory of 𝔰𝔩2 and Uq(𝔰𝔩2). In addition to the classification we make explicit the relationship between bidiagonal pairs and modules for 𝔰𝔩2 and Uq(𝔰𝔩2).

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