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 $T_{h,α}^{j,z}$ matrices for arbitrary ($j, z$) representations of the Jordanian Fun$_{h,α}$(GL(2)) algebra may be extracted from the corresponding $T_{q,λ}^{j,z}$ matrices of the standard Fun$_{q,λ}$(GL(2)) algebra.
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.
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 ${\c…
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.
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…
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.
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.
Classical r-matrices with carriers g_c containing the Borel subalgebra b(+,g) of simple Lie algebra g are studied. Using the graphycal presentation of the dual Lie algebra g#(r) we show that such solutions of the CYBE always exist. To obtain the explicit form of these r-matrices we find the dual coordinates in which the adjoint action of g_c can be reduced. This gives us the detailed structure of the Jordanian r-matrices that are the candidates for enlarging the initial full chain r-matrix. We search the desired solution in the factorized form. This leads to the unique transformation: the init…
We study the quantum plane associated to the coloured quantum group GL_{q}^{λ,μ}(2) and solve the problem of constructing the corresponding differential geometric structure. This is achieved within the R-matrix framework generalising the Wess-Zumino formalism and leads to the concept of coloured quantum space. Both, the coloured Manin plane as well as the bicovariant differential calculus exhibit the colour exchange symmetry. The coloured h-plane corresponding to the coloured Jordanian quantum group GL_{h}^{λ,μ}(2) is also obtained by contraction of the coloured q-plane.
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 $sl(2,\R)$ …
We give an intepretation of the Cremmer-Gervais r-matrices for sl(n) in terms of actions of elements in the rational and trigonometric Cherednik algebras of type GL2 on certain subspaces of their polynomial representations. This is used to compute the nilpotency index of the Jordanian r-matrices, thus answering a question of Gerstenhaber and Giaquinto. We also give an interpretation of the Cremmer-Gervais quantization in terms of the corresponding double affine Hecke algebra.
We discussed quantum deformations of D=4 Lorentz and Poincare algebras. In the case of Poincare algebra it is shown that almost all classical r-matrices of S. Zakrzewski classification correspond to twisted deformations of Abelian and Jordanian types. A part of twists corresponding to the r-matrices of Zakrzewski classification are given in explicit form.
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.
We discussed twisted quantum deformations of D=4 Lorentz and Poincare algebras. In the case of Poincare algebra it is shown that almost all classical r-matrices of S.Zakrzewski classification can be presented as a sum of subordinated r-matrices of Abelian and Jordanian types. Corresponding twists describing quantum deformations are obtained in explicit form. This work is an extended version of the paper \url{arXiv:0704.0081v1 [math.QA]}.
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…
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 …
$GL_h(n) \times GL_h(m)$-covariant $h$-bosonic algebras are built by contracting the $GL_q(n) \times GL_q(m)$-covariant $q$-bosonic algebras considered by the present author some years ago. Their defining relations are written in terms of the corresponding $R_h$-matrices. Whenever $n=2$, and $m=1$ or 2, it is proved by using U_h(sl(2)) Clebsch-Gordan coefficients that they can also be expressed in terms of coupled commutators in a way entirely similar to the classical case. Some U_h(sl(2)) rank-1/2 irreducible tensor operators, recently contructed by Aizawa in terms of standard bosonic operato…
We present a systematic procedure to obtain singular solutions of the constant quantum Yang-Baxter equation in arbitrary dimension. This approach, inspired in the Lie (super)algebra structure, is explicitly applied to the particular case of (graded) contractions of the orthogonal real algebra ${\mathfrak{so}}(N+1)$. In this way we show that "classical" contraction parameters which appear in the commutation relations of the contracted Lie algebras, become quantum deformation parameters, arising as entries of the resulting quantum $R$-matrices.
Given two m x n matrices A = (a_{ij}) and B=(b_{ij}) with entries in B(H), the Schur block product is the m x n matrix A \square B := (a_{ij}b_{ij}). There exists an m x n contraction matrix S = (s_{ij}), such that A \square B = diag(AA*)^(1/2) S diag(B*B)^(1/2). This decomposition is also valid for the block Schur tensor product. It is shown, via the theory of random matrices, that the set of contractions S, which may appear in such a decomposition, is a very thin subset of the unit ball of M_n(B(H)).
$GL_h(n) \times GL_{h'}(m)$-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 …
Chains of extended jordanian twists are studied for the universal enveloping algebras U(so(M)). The carrier subalgebra of a canonical chain F cannot cover the maximal nilpotent subalgebra N(so(M)). We demonstrate that there exist other types of Frobenius subalgebras in so(M) that can be large enough to include N(so(M)). The problem is that the canonical chains F do not preserve the primitivity on these new carrier spaces. We show that this difficulty can be overcome and the primitivity can be restored if one changes the basis and passes to the deformed carrier spaces. Finally the twisting elem…
A universal quasitriangular $R$--matrix for the non-standard quantum (1+1) Poincaré algebra $U_ziso(1,1)$ is deduced by imposing analyticity in the deformation parameter $z$. A family $g_μ$ of ``quantum graded contractions" of the algebra $U_ziso(1,1)\oplus U_{-z}iso(1,1)$ is obtained; this set of quantum algebras contains as Hopf subalgebras with two primitive translations quantum analogues of the two dimensional Euclidean, Poincaré and Galilei algebras enlarged with dilations. Universal $R$--matrices for these quantum Weyl algebras and their associated quantum groups are constructed.
