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 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.

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 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.

scholarly journals k-step nilpotent Lie algebras

2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Michel Goze ◽  
Elisabeth Remm
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Deformation Process ◽  
Nilpotent Lie Algebras ◽  
Early Stages ◽  
Finite Dimensional ◽  
Contraction Process ◽  
Finite Dimensional Lie Algebras

AbstractThe classification of complex or real finite dimensional Lie algebras which are not semi simple is still in its early stages. For example, the nilpotent Lie algebras are classified only up to dimension 7. Moreover, to recognize a given Lie algebra in the classification list is not so easy. In this work, we propose a different approach to this problem. We determine families for some fixed invariants and the classification follows by a deformation process or a contraction process. We focus on the case of 2- and 3-step nilpotent Lie algebras. We describe in both cases a deformation cohomology for this type of algebras and the algebras which are rigid with respect to this cohomology. Other

scholarly journals A categorical reconstruction of crystals and quantum groups at q = 0

2019 ◽  
Vol 70 (3) ◽  
pp. 895-925
Author(s):  
Craig Smith
Keyword(s):  
Hopf Algebra ◽  
Quantum Group ◽  
Lie Algebra ◽  
Quantum Groups ◽  
Analogous Result ◽  
Crystal Bases ◽  
Direct Sums ◽  
Finite Dimensional ◽  
Monoidal Structure

Abstract The quantum co-ordinate algebra Aq(g) associated to a Kac–Moody Lie algebra g forms a Hopf algebra whose comodules are direct sums of finite-dimensional irreducible Uq(g) modules. In this paper, we investigate whether an analogous result is true when q=0. We classify crystal bases as coalgebras over a comonadic functor on the category of pointed sets and encode the monoidal structure of crystals into a bicomonadic structure. In doing this, we prove that there is no coalgebra in the category of pointed sets whose comodules are equivalent to crystal bases. We then construct a bialgebra over Z whose based comodules are equivalent to crystals, which we conjecture is linked to Lusztig’s quantum group at v=∞.

scholarly journals A combinatorial identity for the derivative of a theta series of a finite type root lattice

2003 ◽  
Vol 172 ◽  
pp. 1-30
Author(s):  
Satoshi Naito
Keyword(s):  
Representation Theory ◽  
Lie Algebra ◽  
Finite Type ◽  
Cartan Subalgebra ◽  
Theta Series ◽  
Combinatorial Identity ◽  
Root Lattice ◽  
Simple Lie Algebra ◽  
Finite Dimensional ◽  
Affine Lie Algebra

AbstractLet be a (not necessarily simply laced) finite-dimensional complex simple Lie algebra with the Cartan subalgebra and Q ⊂ * the root lattice. Denote by ΘQ(q) the theta series of the root lattice Q of . We prove a curious “combinatorial” identity for the derivative of ΘQ(q), i.e. for by using the representation theory of an affine Lie algebra.

scholarly journals Supports of weight modules over Witt algebras

2011 ◽  
Vol 141 (1) ◽  
pp. 155-170 ◽  
Author(s):  
Volodymyr Mazorchuk ◽  
Kaiming Zhao
Keyword(s):  
Lie Algebra ◽  
Weight Module ◽  
Root Space ◽  
Abelian Subalgebra ◽  
Space Decomposition ◽  
Finite Dimensional ◽  
Graded Lie Algebra ◽  
Alpha Beta ◽  
Weight Modules

As the first step towards a classification of simple weight modules with finite dimensional weight spaces over Witt algebras Wn, we explicitly describe the supports of such modules. We also obtain some descriptions of the support of an arbitrary simple weight module over a ℤn-graded Lie algebra $\mathfrak{g}$ having a root space decomposition $\smash{\bigoplus_{\alpha\in\mathbb{Z}^n}\mathfrak{g}_\alpha}$ with respect to the abelian subalgebra $\mathfrak{g}_0$, with the property $\smash{[\mathfrak{g}_\alpha,\mathfrak{g}_\beta] = \mathfrak{g}_{\alpha+\beta}}$ for all α, β ∈ ℤn, α ≠ β (this class contains the algebra Wn).

scholarly journals Irreducible representations of simple Lie algebras by differential operators

2021 ◽  
Vol 81 (10) ◽  
Author(s):  
A. Morozov ◽  
M. Reva ◽  
N. Tselousov ◽  
Y. Zenkevich
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Differential Operators ◽  
Irreducible Representations ◽  
Explicit Formulas ◽  
Systematic Method ◽  
First Order ◽  
Highest Weight ◽  
Finite Dimensional ◽  
Lowering Operators

AbstractWe describe a systematic method to construct arbitrary highest-weight modules, including arbitrary finite-dimensional representations, for any finite dimensional simple Lie algebra $${\mathfrak {g}}$$ g . The Lie algebra generators are represented as first order differential operators in $$\frac{1}{2} \left( \dim {\mathfrak {g}} - \text {rank} \, {\mathfrak {g}}\right) $$ 1 2 dim g - rank g variables. All rising generators $$\mathbf{e}$$ e are universal in the sense that they do not depend on representation, the weights enter (in a very simple way) only in the expressions for the lowering operators $$\mathbf{f}$$ f . We present explicit formulas of this kind for the simple root generators of all classical Lie algebras.

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