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On super-Jordanian ${\cal U}_{\sf h}(sl(N|1))$ algebra  [PDF]
B. Abdesselam,A. Chakrabarti,R. Chakrabarti,A. Yanallah,M. B. Zahaf
Mathematics , 2005, DOI: 10.1088/0305-4470/39/26/004
Abstract: A nonlinear realisation of the nonstandard (super-Jordanian) deformed ${\cal U}_{\sf h}(sl(N|1))$ algebra is given for arbitrary $N$.
Irreducible Representations of Jordanian Quantum Algebra ${\cal U}_h(sl(2))$ Via a Nonlinear Map  [PDF]
B. Abdesselam,A. Chakrabarti,R. Chakrabarti
Mathematics , 1996, DOI: 10.1142/S0217732396002861
Abstract: The generators of the Jordanian quantum algebra ${\cal U}_h(sl(2))$ are expressed as nonlinear invertible functions of the classical $sl(2)$ generators. This permits immediate explicit construction of the finite dimensional irreducible representations of the algebra ${\cal U}_h(sl(2))$. Using this construction, new finite dimensional solutions of the Yang-Baxter equation may be obtained.
Super-jordanian deformation of the orthosymplectic Lie superalgebras  [PDF]
P. P. Kulish
Mathematics , 1998,
Abstract: The recently proposed jordanian quantization of the Lie superalgebra $osp(1|2)$ due to the embedding $sl(2) \subset osp(1|2)$, is extended including odd generators into the twisting element $\cal F$. This deformation is obtained as a contraction of the quantum superalgebra ${\cal U}_{q}(osp(1|2))$.
Addendum to "On Super-Jordanian ${\cal U}_{\sf h}(sl(N|1))$ Algebra  [PDF]
B. Abdesselam,A. Chakrabarti,R. Chakrabarti,A. Yanallah,M. B. Zahaf
Mathematics , 2007,
Abstract: We give a complete proof of the result (2.10) presented in our paper published in J. Phys. A: Math. Gen. 39 (2006) 8307.
The Gervais-Neveu-Felder equation for the Jordanian quasi-Hopf U_{h;y}(sl(2)) algebra  [PDF]
A. Chakrabarti,R. Chakrabarti
Mathematics , 2000, DOI: 10.1088/0305-4470/33/25/304
Abstract: Using a contraction procedure, we construct a twist operator that satisfies a shifted cocycle condition, and leads to the Jordanian quasi-Hopf U_{h;y}(sl(2)) algebra. The corresponding universal ${\cal R}_{h}(y)$ matrix obeys a Gervais-Neveu-Felder equation associated with the U_{h;y}(sl(2)) algebra. For a class of representations, the dynamical Yang-Baxter equation may be expressed as a compatibility condition for the algebra of the Lax operators.
Extended jordanian twists for Lie algebras  [PDF]
P. P. Kulish,V. D. Lyakhovsky,A. I. Mudrov
Mathematics , 1998, DOI: 10.1063/1.532987
Abstract: Jordanian quantizations of Lie algebras are studied using the factorizable twists. For a restricted Borel subalgebras ${\bf B}^{\vee}$ of $sl(N)$ the explicit expressions are obtained for the twist element ${\cal F}$, universal ${\cal R}$-matrix and the corresponding canonical element ${\cal T}$. It is shown that the twisted Hopf algebra ${\cal U}_{\cal F} ({\bf B}^{\vee})$ is self dual. The cohomological properties of the involved Lie bialgebras are studied to justify the existence of a contraction from the Dinfeld-Jimbo quantization to the jordanian one. The construction of the twist is generalized to a certain type of inhomogenious Lie algebras.
On Nonstandard Quantizations of osp(2|1) Superalgebra via Contraction and Mapping  [PDF]
B. Abdesselam,R. Chakrabarti,A. Hazzab,A. Yanallah
Mathematics , 2003,
Abstract: We develop a generic reprersentation-independent contraction procedure for obtaining, for instance, $R_{\sf h}$ and $L$ operators of arbitrary dimensions for the quantized ${\cal U}_{\sf h}(osp(2|1))$ algebra corresponding to the classical $r_2$ matrix from the pertinent quantities of the standard q-deformed ${\cal U}_q(osp(2|1))$ algebra. Also the quantized ${\bf U_h}(osp(2|1))$ algebra corresponding to the classical $r_1$ matrix comprising of the generators of the classical $sl(2)$ algebra is obtained in terms of a nonlinear basis set.
Jordanian quantum spheres  [PDF]
R. Chakrabarti,J. Segar
Mathematics , 2001, DOI: 10.1142/S0217732301004832
Abstract: We introduce and investigate a one parameter family of quantum spaces invariant under the left (right) coactions of the group-like element ${\cal T}_{h}^{(j=1)}$ of the Jordanian function algebra $Fun_{h}(SL(2))$. These spaces may be regarded as Jordanian quantization of the two-dimensional spheres.
On Jordanian U_{h,α}(gl(2)) Algebra and Its T Matrices Via a Contraction Method  [PDF]
R. Chakrabarti,C. Quesne
Physics , 1998, DOI: 10.1142/S0217751X9900124X
Abstract: The $R_h^{j_1;j_2}$ matrices of the Jordanian U$_h$(sl(2)) algebra at arbitrary dimensions may be obtained from the corresponding $R_q^{j_1;j_2}$ matrices of the standard $q$-deformed U$_q$(sl(2)) algebra through a contraction technique. By extending this method, the coloured two-parametric ($h, \alpha$) Jordanian $R_{h,\alpha}^{j_1,z_1;j_2,z_2}$ matrices of the U$_{h,\alpha}$(gl(2)) algebra may be derived from the corresponding coloured $R_{q,\lambda}^{j_1,z_1;j_2,z_2}$ matrices of the standard ($q, \lambda$)-deformed U$_{q,\lambda}$(gl(2)) algebra. Moreover, by using the contraction process as a tool, the coloured $T_{h,\alpha}^{j,z}$ matrices for arbitrary ($j, z$) representations of the Jordanian Fun$_{h,\alpha}$(GL(2)) algebra may be extracted from the corresponding $T_{q,\lambda}^{j,z}$ matrices of the standard Fun$_{q,\lambda}$(GL(2)) algebra.
On ${\cal U}_h(sl(2))$, ${\cal U}_h(e(3))$ and their Representations  [PDF]
B. Abdesselam,A. Chakrabarti,R. Chakrabarti
Mathematics , 1996, DOI: 10.1142/S0217751X97001341
Abstract: By solving a set of recursion relations for the matrix elements of the ${\cal U}_h(sl(2))$ generators, the finite dimensional highest weight representations of the algebra were obtained as factor representations. Taking a nonlinear combination of the generators of the two copies of the ${\cal U}_h(sl(2))$ algebra, we obtained ${\cal U}_h(so(4))$ algebra. The latter, on contraction, yields ${\cal U}_h(e(3))$ algebra. A nonlinear map of ${\cal U}_h(e(3))$ algebra on its classical analogue $e(3)$ was obtained. The inverse mapping was found to be singular. It signifies a physically interesting situation, where in the momentum basis, a restricted domain of the eigenvalues of the classical operators is mapped on the whole real domain of the eigenvalues of the deformed operators.
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