Using the contraction procedure introduced by us in Ref. \cite{ACC2}, we construct, in the first part of the present letter, the Jordanian quantum Hopf algebra ${\cal U}_{\sf h}(sl(3))$ which has a remarkably simple coalgebraic structure and contains the Jordanian Hopf algebra ${\cal U}_{\sf h}(sl(2))$, obtained by Ohn, as a subalgebra. A nonlinear map between ${\cal U}_{\sf h}(sl(3))$ and the classical $sl(3)$ algebra is then established. In the second part, we give the higher dimensional Jordanian algebras ${\cal U}_{\sf h}(sl(N))$ for all $N$. The Universal ${\cal R}_{\sf h}$-matrix of ${\cal U}_{\sf h} (sl(N))$ is also given.
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 ${\cal U}_{q}(osp(1|2))$.
It is known that the inhomogeneous quantum group IGL_{q,r}(2) can be constructed as a quotient of the multiparameter q-deformation of GL(3). We show that a similar result holds for the inhomogeneous Jordanian deformation and exhibit its Hopf structure.
Recently, a class of transformations of $R_q$-matrices was introduced such that the $q \to 1$ limit gives explicit nonstandard $R_h$-matrices. The transformation matrix is singular as $q \to 1$. For the transformed matrix, the singularities, however, cancel yielding a well-defined construction. We have shown that our method can be implemented systematically on $R_q$ matrices of all dimensions of $U_q(sl(N)), U_Q(osp(1|2))$ and $U_q(sl(2|1))$ algebras. Explicit constructions are presented for $U_q(sl(2)), U_q(sl(3)), U_q(osp(1|2))$ and $U_q(sl(2|1))$ algebras, while choosing $R_q$ matrix for (f…
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.
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 ($h, α$) Jordanian $R_{h,α}^{j_1,z_1;j_2,z_2}$ matrices of the U$_{h,α}$(gl(2)) algebra may be derived from the corresponding coloured $R_{q,λ}^{j_1,z_1;j_2,z_2}$ matrices of the standard ($q, λ$)-deformed U$_{q,λ}$(gl(2)) algebra. Moreover, by using the contraction process as a tool, the coloured $T_{h,α}^…
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 gene…
We introduce a universal R matrix for the Jordanian deformation of $\U{ \sl(2)}$. Using $\Uh{\so(4)}=\Uh{\sl(2)} \oplus {\rm U}_{-h}(\sl(2))$, we obtain the universal R matrix for $\Uh{\so(4)}$. Applying the graded contractions on the universal R matrix of $\Uh{\so(4)}$, we show that there exist three distinct R matrices for all of the contracted algebras. It is shown that $\Uh{\sl(2)}$, $\Uh{\so(4)}$, and all of these contracted algebras are triangular.
This paper suveys some recent algebraic developments in two parameter Quantum deformations and their Nonstandard (or Jordanian) counterparts. In particular, we discuss the contraction procedure and the quantum group homomorphisms associated to these deformations. The scheme is then set in the wider context of the coloured extensions of these deformations, namely, the so-called Coloured Quantum Groups.
The quantum group GL_p,q(2) is known to be related to the Jordanian GL_h,h'(2) via a contraction procedure. It can also be realised using the generators of the Hopf algebra G_r,s. We contract the G_r,s quantum group to obtain its Jordanian analogue G_m,k, which provides a realisation of GL_h,h'(2) in a manner similar to the q-deformed case.
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.
The quantum group G_r,s provides a realisation of the two parameter quantum GL_p,q(2) which is known to be related to the two parameter nonstandard GL_hh'(2) group via a contraction method. We apply the contraction procedure to G_r,s and obtain a new Jordanian quantum group G_m,k. Furthermore, we provide a realisation of GL_h,h'(2) in terms of G_m,k. The contraction procedure is then extended to the coloured quantum group GL_r{λ,μ}(2) to yield a new Jordanian quantum group GL_m{λ,μ}(2). Both G_r,s and G_m,k are then generalised to their coloured versions which inturn provide similar realisatio…
We consider a superextension of the extended Jordanian twist, describing nonstandard quantization of anti-de-Sitter ($AdS$) superalgebra $osp(1|4)$ in the form of Hopf superalgebra. The super-Jordanian twisting function and corresponding basic coproduct formulae for the generators of $osp(1|4)$ are given in explicit form. The nonlinear transformation of the classical superalgebra basis not modifying the defining algebraic relations but simplifying coproducts and antipodes is proposed. Our physical application is to interpret the new super-Jordanian deformation of $osp(1|4)$ superalgebra as def…
A triangular quantum deformation of $ osp(2/1) $ from the classical $r$-matrix including an odd generator is presented with its full Hopf algebra structure. The deformation maps, twisting element and tensor operators are considered for the deformed $ osp(2/1)$. It is also shown that its subalgebra generated by the Borel subalgebra is self-dual.
We classify all 4-dimensional first order bicovariant calculi on the Jordanian quantum group GL_{h,g}(2) and all 3-dimensional first order bicovariant calculi on the Jordanian quantum group SL_{h}(2). In both cases we assume that the bicovariant bimodules are generated as left modules by the differentials of the quantum group generators. It is found that there are 3 1-parameter families of 4-dimensional bicovariant first order calculi on GL_{h,g}(2) and that there is a single, unique, 3-dimensional bicovariant calculus on SL_{h}(2). This 3-dimensional calculus may be obtained through a classic…
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.
A Gelfan'd--Dyson mapping is used to generate a one-boson realization for the non-standard quantum deformation of $sl(2,\R)$ which directly provides its infinite and finite dimensional irreducible representations. Tensor product decompositions are worked out for some examples. Relations between contraction methods and boson realizations are also explored in several contexts. So, a class of two-boson representations for the non-standard deformation of $sl(2,\R)$ is introduced and contracted to the non-standard quantum (1+1) Poincaré representations. Likewise, a quantum extended Hopf $sl(2,\R)$ …
Starting from the classical r-matrix of the non-standard (or Jordanian) quantum deformation of the sl(2,R) algebra, new triangular quantum deformations for the real Lie algebras so(2,2), so(3,1) and iso(2,1) are simultaneously constructed by using a graded contraction scheme; these are realized as deformations of conformal algebras of (1+1)-dimensional spacetimes. Time-type and space-type quantum algebras are considered according to the generator that remains primitive after deformation: either the time or the space translation, respectively. Furthermore by introducing differential-difference …
We examine certain nonassociative deformations of quantum mechanics and gravity in three dimensions related to the dynamics of electrons in uniform distributions of magnetic charge. We describe a quantitative framework for nonassociative quantum mechanics in this setting, which exhibits new effects compared to ordinary quantum mechanics with sourceless magnetic fields, and the extent to which these theoretical consequences may be experimentally testable. We relate this theory to noncommutative Jordanian quantum mechanics, and show that its underlying algebra can be obtained as a contraction of…
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.
The universal ${\cal R}$--matrix for a quantized Poincar{é} algebra ${\cal P}(3+1)$ introduced by Ballesteros et al is evaluated. The solution is obtained as a specific case of a formulated multidimensional generalization to the non-standard (Jordanian) quantization of $sl(2)$.
We re-examine all the contractions related with the ${\cal U}_q(su(2))$ deformed algebra and study the consequences that the contraction process has for their structure. We also show using ${\cal U}_q(su(2))\times{\cal U}(u(1))$ as an example that, as in the undeformed case, the contraction may generate Hopf algebra cohomology. We shall show that most of the different Hopf algebra deformations obtained have a bicrossproduct or a cocycle bicrossproduct structure, for which we shall also give their dual `group' versions. The bicovariant differential calculi on the deformed spaces associated with…
We study the biparametric quantum deformation of GL(2) x GL(1) and exhibit its cross-product structure. We derive explictly the associated dual algebra, i.e., the quantised universal enveloping algebra employing the R-matrix procedure. This facilitates construction of a bicovariant differential calculus which is also shown to have a cross-product structure. Finally, a Jordanian analogue of the deformation is presented as a cross-product algebra.
We investigate the algebro-geometric structure of a novel two-parameter quantum deformation which exhibits the nature of a semidirect or cross-product algebra built upon GL(2) x GL(1), and is related to several other known examples of quantum groups. Following the R-matrix framework, we construct the L+/- functionals and address the problem of duality for this quantum group. This naturally leads to the construction of a bicovariant differential calculus that depends only on one deformation parameter, respects the cross-product structure and has interesting applications. The corresponding Jorda…
Polynomials of boson creation and annihilation operators which form irreducible tensor operators for Jordanian quantum algebra U_h(sl(2)), called h-symplecton, are introduced and their properties are investigated. It is shown that many properties of symplecton for Lie algebra sl(2) are extended to h-symplecton. The h-symplecton is also a basis of irreducible representation of SL_h(2) dual to U_h(sl(2)). As an application of the procedure used to construct h-symplecton, we construct the representation bases of SL_h(2) on the quantum h-plane.
We show how some classical r-matrices for the D=4 Poincare algebra can be supersymmetrized by an addition of part depending on odd supercharges. These r-matrices for D=4 super-Poincare algebra can be presented as a sum of the so-called subordinated r-matrices of super-Abelian and super-Jordanian type. Corresponding twists describing quantum deformations are obtained in an explicit form. These twists are the super-extensions of twists obtained in the paper arXiv:0712.3962.
Spaces of constant curvature and their motion groups are described most naturally in Cartesian basis. All these motion groups also known as CK groups are obtained from orthogonal group by contractions and analytical continuations. On the other hand quantum deformation of orthogonal group $SO(N) $ is most easily performed in so-called symplectic basis. We reformulate its standard quantum deformation to Cartesian basis and obtain all possible contractions of quantum orthogonal group $SO_q(N)$ both for untouched and transformed deformation parameter. It turned out, that similar to undeformed case…
A class of transformations of $R_q$-matrices is introduced such that the $q\to 1$ limit gives explicit nonstandard $R_{h}$-matrices. The transformation matrix is singular itself at $q\to 1$ limit. For the transformed matrix, the singularities, however, cancel yielding a well-defined construction. Our method can be implemented systematically for R-matrices of all dimensions and not only for $sl(2)$ but also for algebras of higher dimensions. Explicit constructions are presented starting with ${\cal U}_q(sl(2))$ and ${\cal U}_q(sl(3))$, while choosing $R_q$ for (fund. rep.)$\otimes$(arbitrary ir…
We show that if $g_Γ$ is the quantum tangent space (or quantum Lie algebra in the sense of Woronowicz) of a bicovariant first order differential calculus over a coquasitriangular Hopf algebra $(A,r)$, then a certain extension of it is a braided Lie algebra in the category of $A$-comodules. This is used to show that the Woronowicz quantum universal enveloping algebra $U(g_Γ)$ is a bialgebra in the braided category of $A$-comodules. We show that this algebra is quadratic when the calculus is inner. Examples with this unexpected property include finite groups and quantum groups with their standar…
$GL_h(n) \times GL_{h'}(m)$-covariant (hh')-bosonic (or (hh')-fermionic) algebras ${\cal A}_{hh'\pm}(n,m)$ are built in terms of the corresponding R_h and $R_{h'}$-matrices by contracting the $GL_q(n) \times GL_{q^{\pm1}}(m)$-covariant q-bosonic (or q-fermionic) algebras ${\cal A}^{(α)}_{q\pm}(n,m)$, $α= 1, 2$. When using a basis of ${\cal A}^{(α)}_{q\pm}(n,m)$ wherein the annihilation operators are contragredient to the creation ones, this contraction procedure can be carried out for any n, m values. When employing instead a basis wherein the annihilation operators, as the creation ones, are …
