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.
A two-parametric non-standard (Jordanian) deformation of the Lie algebra $gl(2)$ is constructed, and then, exploited to obtain a new, triangular R-matrix solution of the coloured Yang-Baxter equation. The corresponding coloured quantum group is presented explicitly.
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}^…
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 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.
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.
We discuss quantum deformations of Jordanian type for Lie superalgebras. These deformations are described by twisting functions with support from Borel subalgebras and they are multiparameter in the general case. The total twists are presented in explicit form for the Lie superalgebras sl(m|n) and osp(1|2n). We show also that the classical $r$-matrix for a light-cone deformation of D=4 super-Poincare algebra is of Jordanian type and a corresponding twist is given in explicit form.
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…
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 ${\cal U}_{q}(osp(1|2))$.
The general solution to the reflection equation associated with the jordanian deformation of the SL(2) invariant Yang R-matrix is found. The same K-matrix is obtained by the special scaling limit of the XXZ-model with general boundary conditions. The Hamiltonian with the boundary terms is explicitly derived according to the Sklyanin formalism. We discuss the structure of the spectrum of the deformed XXX-model and its dependence on the boundary conditions.
We derive the deformed sl(2) Gaudin model with integrable boundaries. Starting from the Jordanian deformation of the SL(2)-invariant Yang R-matrix and generic solutions of the associated reflection equation and the dual reflection equation, the corresponding inhomogeneous spin-1/2 XXX chain is obtained. The quasi-classical expansion of the transfer matrix yields the deformed sl(2) Gaudin Hamiltonians with boundary terms.
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.
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.
Representation theory for the Jordanian quantum algebra U=U_h(sl(2)) is developed using a nonlinear relation between its generators and those of sl(2). Closed form expressions are given for the action of the generators of U on the basis vectors of finite dimensional irreducible representations. In the tensor product of two such representations, a new basis is constructed on which the generators of U have a simple action. Using this basis, a general formula is obtained for the Clebsch-Gordan coefficients of U. It is shown that the Clebsch-Gordan matrix is essentially the product of a triangular…
By the supersymmetrization of a simple algebraic technique proposed in \cite{LuTo2017} we obtain the complete classification of all basic (nonisomorphic) quantum deformations for the orthosymplectic Lie superalgebra $\mathfrak{osp}(1|2;\mathbb{C})$ and its pseudoreal and real forms in terms of the classical $r$-matrices. In particular, we prove that pseudoreal compact form has only one quantum deformation (standart $q$-analog), and the $D=3$, $N=1$ Lorentz supersymmetry, which is the non-compact real form of $\mathfrak{osp}(1|2;\mathbb{C})$, has four different Hopf-algebraic quantum deformatio…
The limiting transitions between different types of quantizations are studied by the deformation theory methods. We prove that for the first order coboundary deformation (g,g*_1 + x g*_2) of a Lie bialgebra (g,g*) one can always get the quantized Lie bialgebra (g,g*_2) as a limit of the sequence of quantizations of the type A(g,g*_1). The obtained results are illustrated by some low-dimensional examples of quantized Lie algebras and superalgebras.
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.
A non-standard quantum deformation of the two-photon algebra $h_6$ is constructed, and its quantum universal R-matrix is given. Representations of this new quantum algebra are studied on the Fock space and translated into Fock-Bargmann realizations that provide a direct formalism for the definition of deformed states of light. Finally, the isomorphism between $h_6$ and the (1+1) Schrödinger algebra is used to introduce a new (non-standard) Hopf algebra deformation of this latter symmetry algebra.
The quantum deformation of the Jordanian twist F_qJ for the standard quantum Borel algebra U_q(B) is constructed. It gives the family U_qJ(B) of quantum algebras depending on parameters x and h. In a generic point these algebras represent the hybrid (standard-nonstandard) quantization. The quantum Jordanian twist can be applied to the standard quantization of any Kac-Moody algebra. The corresponding classical r-matrix is a linear combination of the Drinfeld- Jimbo and the Jordanian ones. The obtained two-parametric families of Hopf algebras are smooth and for the limit values of the parameters…
Twist deformation U_F(g) is equivalent to the quantum group Fun_d(G#) and has two preferred bases: the one originating from U(g) and that of the coordinate functions on the dual Lie group G#. The costructure of the Hopf algebra U_F(g) is analized in terms of group G#. The weight diagram of the adjoint representation of the algebra g# is constructed in terms of the root system L(g). The explicit form of the g --> g# transformation can be obtained for any simple Lie algebra g and the factorizable chain F of extended Jordanian twists. The dual group approach is used to find new solutions of the t…
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 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…
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]}.
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.
This paper together with the previous one (arXiv:hep-th/0604146) presents the detailed description of all quantum deformations of D=4 Lorentz algebra as Hopf algebra in terms of complex and real generators. We describe here in detail two quantum deformations of the D=4 Lorentz algebra o(3,1) obtained by twisting of the standard q-deformation U_{q}(o(3,1)). For the first twisted q-deformation an Abelian twist depending on Cartan generators of o(3,1) is used. The second example of twisting provides a quantum deformation of Cremmer-Gervais type for the Lorentz algebra. For completeness we describ…
The twisting function describing a nonstandard (super-Jordanian) quantum deformation of $osp(1|2)$ is given in explicite closed form. The quantum coproducts and universal R-matrix are presented. The non-uniqueness of the twisting function as well as two real forms of the deformed $osp(1|2)$ superalgebras are considered. One real quantum $osp(1|2)$ superalgebra is interpreted as describing the $κ$-deformation of D=1, N=1 superconformal algebra, which can be applied as a symmetry algebra of N=1 superconformal mechanics.
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…
Drinfeld twist is applied to the Lie algebra gl(2) so that a two-parametric deformation of it is obtained, which is identical to the Jordanian deformation of the gl(2) obtained by Aneva et al. The same twist element is applied to deform the Lie superalgebra sl(1/2), since the gl(2) is embedded into the sl(1/2). By making use of the FRT-formalism, we construct a deformation of the Lie supergroup $ SL(1/2). $
We define a deformation of our earlier link homologies for fundamental representations of sl_m. The deformed homology of a link is isomorphic to the deformed homology of the disjoint union of its components. Moreover, there exists a spectral sequence starting with the old homology and converging to this deformed homology.
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…
The Drinfeld twist is applyed to deforme the rank one orthosymplectic Lie superalgebra $osp(1|2)$. The twist element is the same as for the $sl(2)$ Lie algebra due to the embedding of the $sl(2)$ into the superalgebra $osp(1|2)$. The R-matrix has the direct sum structure in the irreducible representations of $osp(1|2)$. The dual quantum group is defined using the FRT-formalism. It includes the Jordanian quantum group $SL_ξ(2)$ as subalgebra and Grassmann generators as well.
A novel Hopf algebra $ ( {\tilde G}_{r,s} )$, depending on two deformation parameters and five generators, has been constructed. This $ {\tilde G}_{r,s}$ Hopf algebra might be considered as some quantisation of classical $GL(2) \otimes GL(1) $ group, which contains the standard $GL_q(2)$ quantum group (with $ q=r^{-1} $) as a Hopf subalgebra. However, we interestingly observe that the two parameter deformed $GL_{p,q}(2)$ quantum group can also be realised through the generators of this $ {\tilde G}_{r,s}$ algebra, provided the sets of deformation parameters $p,~q$ and $r,~s$ are related to eac…
We propose in this thesis a new deformation process of Kac-Moody algebras and their representations. The direction of deformation is given by a collection of numbers, called a colouring. The natural numbers lead for example to the classical algebras, while the quantum numbers lead to the associated quantum algebras. We first establish sufficient and necessary conditions on colourings to allow the process depend polynomially on a formal parameter and to provide the generalised quantum enveloping (GQE) algebras. We then lift the restrictions and show that the process still exists via the coloure…
New deformations of the Poincare group $Fun(P(1+1))$ and its dual enveloping algebra $U(p(1+1))$ are obtained as a contraction of the $h$-deformed (Jordanian) quantum group $Fun(SL_h(2))$ and its dual. A nonstandard quantization of the Heisenberg algebra $U(h(1))$ is also investigated.
We extend the notion of dually conjugate Hopf (super)algebras to the coloured Hopf (super)algebras ${\cal H}^c$ that we recently introduced. We show that if the standard Hopf (super)algebras ${\cal H}_q$ that are the building blocks of ${\cal H}^c$ have Hopf duals ${\cal H}_q^*$, then the latter may be used to construct coloured Hopf duals ${\cal H}^{c*}$, endowed with coloured algebra and antipode maps, but with a standard coalgebraic structure. Next, we review the case where the ${\cal H}_q$'s are quantum universal enveloping algebras of Lie (super)algebras $U_q(g)$, so that the correspondin…
A universal $R$-matrix for the non-standard (Jordanian) quantum deformation of $sl(2,\R)$ is presented. A family of solutions of the quantum Yang--Baxter equation is obtained from some finite dimensional representations of this Lie bialgebra quantization of $sl(2,\R)$.
The minimal twist map introduced by B. Abdesselam, A. Chakrabarti, R. Chakrabarti and J. Segar (Mod. Phys. Lett. A 14 (1999) 765) for the non-standard (Jordanian) quantum sl(2,R) algebra is used to construct the twist maps for two different non-standard quantum deformations of the (1+1) Schrodinger algebra. Such deformations are, respectively, the symmetry algebras of a space and a time uniform lattice discretization of the (1+1) free Schrodinger equation. It is shown that the corresponding twist maps connect the usual Lie symmetry approach to these discrete equations with non-standard quantum…
The non-standard (Jordanian) quantum deformations of $so(2,2)$ and (2+1) Poincaré algebras are constructed by starting from a quantum $sl(2,\R)$ basis such that simple factorized expressions for their corresponding universal $R$-matrices are obtained. As an application, the null-plane quantum (2+1) Poincaré Poisson-Lie group is quantized by following the FRT prescription. Matrix and differential representations of this null-plane deformation are presented, and the influence of the choice of the basis in the resultant $q$-Schrödinger equation governing the deformed null plane evolution is comme…
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 …
Inspired by Reshetikhin's twisting procedure to obtain multiparametric extensions of a Hopf algebra, a general `symmetry transformation' of the `particle conserving' $R$-matrix is found such that the resulting multiparametric $R$-matrix, with a spectral parameter as well as a colour parameter, is also a solution of the Yang-Baxter equation (YBE). The corresponding transformation of the quantum YBE reveals a new relation between the associated quantized algebra and its multiparametric deformation. As applications of this general relation to some particular cases, multiparametric and coloured ex…
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 $sl(2,\R)$ …
A two-parametric generalization of the Jordanian deformation $U_h (sl(2))$ of $sl(2)$ is presented. This involves Jacobian elliptic functions. In our deformation $U_{(h,k)}(sl(2))$, for $k^2=1$ one gets back $U_h(sl(2))$. The constuction is presented via a nonlinear map on $sl(2)$. This invertible map directly furnishes the highest weight irreducible representations of $U_{(h,k)}(sl(2))$. This map also provides two distinct induced Hopf stuctures, which are exhibited. One is induced by the classical $sl(2)$ and the other by the distinct one of $U_h(sl(2))$. Automorphisms related to the two per…
We propose a general procedure to construct noncommutative deformations of an algebraic submanifold $M$ of $\mathbb{R}^n$, specializing the procedure [G. Fiore, T. Weber, Twisted submanifolds of $\mathbb{R}^n$, arXiv:2003.03854] valid for smooth submanifolds. We use the framework of twisted differential geometry of Aschieri et al. (Class. Quantum Grav. 23, 1883-1911, 2006), whereby the commutative pointwise product is replaced by the $\star$-product determined by a Drinfel'd twist. We actually simultaneously construct noncommutative deformations of all the algebraic submanifolds $M_c$ that are…
We introduce a parametrization of formal deformations of Verma modules of $\mathfrak{sl}_2$. A point in the moduli space is called a colouring. We prove that for each colouring $ψ$ satisfying a regularity condition, there is a formal deformation $U_h(ψ)$ of $U(\mathfrak{sl}_2)$ acting on the deformed Verma modules. We retrieve in particular the quantum algebra $U_h(\mathfrak{sl}_2)$ from a colouring by $q$-numbers. More generally, we establish that regular colourings parametrize a broad family of formal deformations of the Chevalley-Serre presentation of $U(\mathfrak{sl}_2)$. The present paper…
On basis of generalized 6j-symbols we give a formulation of topological quantum field theories for 3-manifolds including observables in the form of coloured graphs. It is shown that the 6j-symbols associated with deformations of the classical groups at simple even roots of unity provide examples of this construction. Calculational methods are developed which, in particular, yield the dimensions of the state spaces as well as a proof of the relation, previously announced for the case of $SU_q(2)$ by V.Turaev, between these models and corresponding ones based on the ribbon graph construction of …
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…
