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 Uq(A1(1))U_q(A_{1}^{(1)}) and Uq(C(2)(2))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 Uq(A2(2))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 Uq(sl^2)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