167 research outputs found

### Central Extension of the Yangian Double

Central extension \DYg of the Double of the Yangian is defined for a simple
Lie algebra ${\bf g}$ with complete proof for ${\bf g} =sl_2$. Basic
representations and intertwining operators are constructed for \DY2.Comment: 12 pages, latex, no figure

### Unitarity and Complete Reducibility of Certain Modules over Quantized Affine Lie Algebras

Let $U_q(\hat{\cal G})$ denote the quantized affine Lie algebra and
$U_q({\cal G}^{(1)})$ the quantized {\em nontwisted} affine Lie algebra. Let
${\cal O}_{\rm fin}$ be the category defined in section 3. We show that when
the deformation parameter $q$ is not a root of unit all integrable
representations of $U_q(\hat{\cal G})$ in the category ${\cal O}_{\rm fin}$ are
completely reducible and that every integrable irreducible highest weight
module over $U_q({\cal G}^{(1)})$ corresponding to $q>0$ is equivalent to a
unitary module.Comment: 17 pages (minor errors corrected

### Unified description of quantum affine (super)algebras U_q(A_{1}^{(1)}) and U_q(C(2)^{(2)})

We show that the quantum affine algebra U_{q}(A_{1}^{(1)}) and the quantum
affine superalgebra U_{q}(C(2)^{(2)}) admit unified description. The difference
between them consists in the phase factor which is equal to 1 for
U_{q}(A_{1}^{(1)}) and is equal to -1 for U_{q}(C(2)^{(2)}). We present such a
description for the construction of Cartan-Weyl generators and their
commutation relations, as well for the universal R-matrices.Comment: 16 pages, LaTeX. Talk by V.N. Tolstoy at XIV-th Max Born Symposium
"New Symmetries and Integrable Models", Karpacz, September 1999; in press in
Proceedings, Ed. World Scientific, 200

### Quantum Affine (Super)Algebras $U_q(A_{1}^{(1)})$ and $U_q(C(2)^{(2)})$

We show that the quantum affine algebra $U_{q}(A_{1}^{(1)})$ and the quantum
affine superalgebra $U_{q}(C(2)^{(2)})$ admit a unified description. The
difference between them consists in the phase factor which is equal to 1 for
$U_{q}(A_{1}^{(1)})$ and it is equal to -1 for $U_{q}(C(2)^{(2)})$. We present
such a description for the actions of the braid group, for the construction of
Cartan-Weyl generators and their commutation relations, as well for the
extremal projector and the universal R-matrix. We give also a unified
description for the 'new realizations' of these algebras together with explicit
calculations of corresponding R-matrices.Comment: 22 pages, LaTe

### Three realizations of quantum affine algebra $U_q(A_2^{(2)})$

In this article we establish explicit isomorphisms between three realizations
of quantum twisted affine algebra $U_q(A_2^{(2)})$: the Drinfeld ("current")
realization, the Chevalley realization and the so-called $RLL$ realization,
investigated by Faddeev, Reshetikhin and Takhtajan.Comment: 15 page

### Cherednik algebras and Zhelobenko operators

We study canonical intertwining operators between induced modules of the trigonometric Cherednik algebra. We demonstrate that these operators correspond to the Zhelobenko operators for the affine Lie algebra of type A. To establish the correspondence, we use the functor of Arakawa, Suzuki and Tsuchiya which maps certain modules of the affine Lie algebra to modules of the Cherednik algebra

### Q-power function over Q-commuting variables and deformed XXX, XXZ chains

We find certain functional identities for the Gauss q-power function of a sum
of q-commuting variables. Then we use these identities to obtain two-parameter
twists of the quantum affine algebra U_q (\hat{sl}_2) and of the Yangian
Y(sl_2). We determine the corresponding deformed trigonometric and rational
quantum R-matrices, which then are used in the computation of deformed XXX and
XXZ Hamiltonians.Comment: LaTeX, 12 page

### Factorization of the Universal R-matrix for $U_q(\hat{sl}_2)$

The factorization of the universal R-matrix corresponding to so called
Drinfeld Hopf structure is described on the example of quantum affine algebra
$U_q(\hat{sl}_2)$. As a result of factorization procedure we deduce certain
differential equations on the factors of the universal ${\cal R}$-matrix, which
allow to construct uniquely these factors in the integral form.Comment: 28 pages, LaTeX 2.09 using amssym.def and amssym.te

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