Gauss decomposition of trigonometric R-matrices

The general formula for the universal R-matrix for quantized nontwisted affine algebras by Khoroshkin and Tolstoy is applied for zero central charge highest weight modules of the quantized affine algebras. It is shown how the universal R-matrix produces the Gauss decomposition of trigonomitric R-matrix in tensor product of these modules. Explicit calculations for the simplest case of $A_1^{(1)}$ are presented. As a consequence new formulas for the trigonometric R-matrix with a parameter in tensor product of $U_q(sl_2)$-Verma modules are obtained.Comment: 14 page

Quantum deformations of D=4 Euclidean, Lorentz, Kleinian and quaternionic o^*(4) symmetries in unified o(4;C) setting

We employ new calculational technique and present complete list of classical $r$-matrices for $D=4$ complex homogeneous orthogonal Lie algebra $\mathfrak{o}(4;\mathbb{C})$, the rotational symmetry of four-dimensional complex space-time. Further applying reality conditions we obtain the classical $r$-matrices for all possible real forms of $\mathfrak{o}(4;\mathbb{C})$: Euclidean $\mathfrak{o}(4)$, Lorentz $\mathfrak{o}(3,1)$, Kleinian $\mathfrak{o}(2,2)$ and quaternionic $\mathfrak{o}^{\star}(4)$ Lie algebras. For $\mathfrak{o}(3,1)$ we get known four classical $D=4$ Lorentz $r$-matrices, but for other real Lie algebras (Euclidean, Kleinian, quaternionic) we provide new results and mention some applications.Comment: 13 pages; typos corrected. v3 matches version published in PL

Quantum deformations of D=4D=4 Euclidean, Lorentz, Kleinian and quaternionic o⋆(4)\mathfrak{o}^{\star}(4) symmetries in unified o(4;C)\mathfrak{o}(4;\mathbb{C}) setting -- Addendum

In our previous paper we obtained a full classification of nonequivalent quasitriangular quantum deformations for the complex $D=4$ Euclidean Lie symmetry $\mathfrak{o}(4;\mathbb{C})$. The result was presented in the form of a list consisting of three three-parameter, one two-parameter and one one-parameter nonisomorphic classical $r$-matrices which provide 'directions' of the nonequivalent quantizations of $\mathfrak{o}(4;\mathbb{C})$. Applying reality conditions to the complex $\mathfrak{o}(4;\mathbb{C})$ $r$-matrices we obtained the nonisomorphic classical $r$-matrices for all possible real forms of $\mathfrak{o}(4;\mathbb{C})$: Euclidean $\mathfrak{o}(4)$, Lorentz $\mathfrak{o}(3,1)$, Kleinian $\mathfrak{o}(2,2)$ and quaternionic $\mathfrak{o}^{\star}(4)$ Lie algebras. In the case of $\mathfrak{o}(4)$ and $\mathfrak{o}(3,1)$ real symmetries these $r$-matrices give the full classifications of the inequivalent quasitriangular quantum deformations, however for $\mathfrak{o}(2,2)$ and $\mathfrak{o}^{\star}(4)$ the classifications are not full. In this paper we complete these classifications by adding three new three-parameter $\mathfrak{o}(2,2)$-real $r$-matrices and one new three-parameter $\mathfrak{o}^{\star}(4)$-real $r$-matrix. All nonisomorphic classical $r$-matrices for all real forms of $\mathfrak{o}(4;\mathbb{C})$ are presented in the explicite form what is convenient for providing the quantizations. We will mention also some applications of our results to the deformations of space-time symmetries and string $\sigma$-models.Comment: 10 pages. We supplement results of our previous paper by adding new $\mathfrak{o}(2,2)$ and $\mathfrak{o}^{\star}(4)$ $r$-matrices needed for the complete classification of real classical $r$-matrices for all four real forms of $\mathfrak{o}(4;\mathbb{C})

Modified Affine Hecke Algebras and Drinfeldians of Type A

We introduce a modified affine Hecke algebra \h{H}^{+}_{q\eta}({l}) (\h{H}_{q\eta}({l})) which depends on two deformation parameters $q$ and $\eta$. When the parameter $\eta$ is equal to zero the algebra \h{H}_{q\eta=0}(l) coincides with the usual affine Hecke algebra \h{H}_{q}(l) of type $A_{l-1}$, if the parameter q goes to 1 the algebra \h{H}^{+}_{q=1\eta}(l) is isomorphic to the degenerate affine Hecke algebra \Lm_{\eta}(l) introduced by Drinfeld. We construct a functor from a category of representations of $H_{q\eta}^{+}(l)$ into a category of representations of Drinfeldian $D_{q\eta}(sl(n+1))$ which has been introduced by the first author.Comment: 11 pages, LATEX. Contribution to Proceedings "Quantum Theory and Symmetries" (Goslar, July 18-22, 1999) (World Scientific, 2000

Extremal projectors for contragredient Lie (super)symmetries (short review)

A brief review of the extremal projectors for contragredient Lie (super)symmetries (finite-dimensional simple Lie algebras, basic classical Lie superalgebras, infinite-dimensional affine Kac-Moody algebras and superalgebras, as well as their quantum $q$-analogs) is given. Some bibliographic comments on the applications of extremal projectors are presented.Comment: 21 pages, LaTeX; typos corrected, references adde

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