An elliptic two-parameter deformation of the (universal enveloping superalgebra of) affine Lie superalgebra $osp(1|2)^{(1)}$ is proposed in terms of free boson realization. This deformed superalgebra is shown to fit in the framework of infinite Hopf family of superalgebras, a generalization of the infinite Hopf family of algebras proposed earlier by the authors. The trigonometric and rational degenerations are briefly discussed.
The complete set of formulas describing the new quantum deformation of the OSP(1;2) supergroup is provided. A general ansatz is solved for the deformation of the Borel subalgebra of its dual quantum deformation of osp(1;2).
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 ${\cal U}_{q}(osp(1|2))$.
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.
In this paper Witten type deformation of osp(1/2) algbera is introduced and its realization and matrix representation are obtained. The matrix representation is shown to be possible only when the dimension is odd.
We analyse the Witten-Woronowicz's type deformations of the Lie superalgebra osp(2,2) and obtain a deformation parametrized by three independent parameters. For some of these algebras, finite dimensional representations are formulated in terms of a finite difference operator, providing operators that are relevant for the classification of quasi exactly solvable, finite difference equations. Similar representations are also pointed out for the Lie superalgebra osp(1,2).
Triangular deformation of the orthosymplectic Lie superalgebra osp(1|4) is defined by chains of twists. Corresponding classical r-matrix is obtained by a contraction procedure from the trigonometric r-matrix. The carrier space of the constant r-matrix is the Borel subalgebra.
The paper contains essentially two new results. Physically, a deformation of the parastatistics in a sense of quantum groups is carried out. Mathematically, an alternative to the Chevalley description of the quantum orthosymplectic superalgebra U_q[osp(2n+1/2m)] in terms of $m$ pairs of deformed parabosons and $n$ pairs of deformed parafermions is outlined.
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.
A two-parameter quantum deformation of the affine Lie super algebra $osp(2|2)^{(2)}$ is introduced and studied in some detail. This algebra is the first example associated with nonsimply-laced and twisted root systems of a quantum current algebra with the structure of a so-called infinite Hopf family of (super)algebras. A representation of this algebra at $c=1$ is realized in the product Fock space of two commuting sets of Heisenberg algebras.
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.
We explicitly compute the first and second cohomology groups of the classical Lie superalgebras $sl_{m|n}$ and $osp_{2|2n}$ with coefficients in the finite dimensional irreducible modules and the Kac modules. We also show that the second cohomology groups of these Lie superalgebras with coefficients in the respective universal enveloping algebras (under the adjoint action) vanish. The latter result in particular implies that the universal enveloping algebras $U(sl_{m|n})$ and $U(osp_{2|2n})$ do not admit any non-trivial formal deformations of Gerstenhaber type.
In [ZH], R.B. Zhang found a way to link certain formal deformations of the Lie algebra o(2l+1) and the Lie superalgebra osp(1,2l). The aim of this article is to reformulate the Zhang transformation in the context of the quantum enveloping algebras "à la" Drinfeld-Jimbo and to show how this construction can explain the main theorem of [GL2]: the annihilator of a Verma module over the Lie superalgebra osp(1,2l) is generated by its intersection with the centralizer of the even part of the enveloping algbra.
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 Racah problem for the quantum superalgebra $\mathfrak{osp}_{q}(1|2)$ is considered. The intermediate Casimir operators are shown to realize a $q$-deformation of the Bannai-Ito algebra. The Racah coefficients of $\mathfrak{osp}_q(1|2)$ are calculated explicitly in terms of basic orthogonal polynomials that $q$-generalize the Bannai-Ito polynomials. The relation between these $q$-deformed Bannai-Ito polynomials and the $q$-Racah/Askey-Wilson polynomials is discussed.
We study the center of the two-parameter quantum supergroup $U_{r,s}(osp(1,2))$ using the Dickson polynomial. We show that the Scasimir operator is completely determined by the $q$-deformed Chebychev polynomial, generalizing an earlier work of Arnaudon and Bauer.
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 type-I simple Lie-superalgebras are $sl(m|n)$ and $osp(2|2n)$. We study the quantum deformations of their untwisted affine extensions $U_q(sl(m|n)^{(1)})$ and $U_q(osp(2|2n)^{(1)})$. We identify additional relations between the simple generators (``extra $q$-Serre relations") which need to be imposed to properly define $\uqgh$ and $U_q(osp(2|2n)^{(1)})$. We present a general technique for deriving the spectral parameter dependent R-matrices from quantum affine superalgebras. We determine the R-matrices for the type-I affine superalgebra $U_q(sl(m|n)^{(1)})$ in various representations, ther…
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.
New solutions of the quantum Yang-Baxter equation, depending in general on three arbitrary parameters, are written down. They are based on the root of unity representations of the quantum orthosymplectic superalgebra \\U, which were found recently. Representations of the braid group $B_N$ are defined within any $N^{th}$ tensorial power of root of 1 \\U modules.
The space of symbols of differential operators on a smooth manifold (i.e., the space of symmetric contravariant tensor fields) is naturally a module over the Lie algebra of vector fields. We study, in the case of $\bf R^n$ with $n\geq2$, multi-parameter formal deformations of this module. The space of linear differential operators on $\bf R^n$ provides an important class of such formal deformations; we show, however, that the whole space of deformations is much larger.
We consider Poisson superalgebras with constant nondegenerate bracket realized on the smooth Grassmann-valued functions with compact supports in R^{2n}. The deformations with even and odd deformation parameters of these superalgebras are presented for n>1.
We discuss two-parameter deformations of an universal enveloping algebra $U(g[u])$ of a polynomial loop algebra $g[u]$, where $g$ is a finite-dimensional complex simple Lie algebra (or superalgebra). These deformations are Hopf algebras. One deformation called Drinfeldian is a quantization of $U(g[u])$ in the direction of a classical r-matrix which is a sum of the simplest rational and trigonometric r-matrices. Another deformation (discussed only for the case $g=sl_{2}$) is a twisting of the usual Yangian $Y_η(sl_{2})$.
We introduce a two-parameter deformation of 2x2 matrices without imposing any condition on the matrices and give the universal R-matrix of the nonstandard quantum group which satisfies the quantum Yang-Baxter relation. Although in the standard two-parameter deformation the quantum determinant is not central, in the nonstandard case it is central. We note that the quantum group thus obtained is related to the quantum supergroup $GL_{p,q}(1|1)$ by a transformation.
We consider new class of classical r-matrices for D=3 and D=4 conformal Lie algebras. These r-matrices do satisfy the classical Yang-Baxter equation and as two-tensors belong to the tensor product of Borel subalgebra. In such a way we generalize the lowest order of known nonstandard quantum deformation of sl(2) to the Lie algebras sp(4) = so(5) and sl(4) = so(6). As an exercise we interpret nonstandard deformation of sl(2) as describing quantum D=1 conformal algebra with fundamental mass parameter. Further we describe the D=3 and D=4 conformal bialgebras with deformation parameters equal to th…
We define a three-parameter deformation of the Weyl-Heisenberg algebra that generalizes the q-oscillator algebra. By a purely algebraical procedure, we set up on this quantum space two differential calculi that are shown to be invariant on the same quantum group, extended to a ten-generator Hopf-star-algebra. We prove that, when the values of the parameters are related, the two differential calculi reduce to one that is invariant under two quantum groups.
We analyse the Witten-Woronowicz's type deformations of the Lie superalgebras spl(N,1) and obtain deformations parametrized by N(N-1)/2 independent parameters for N>2 and by three parameters for N=2. For some of these algebras, finite dimensional representations can be constructed in terms of finite difference operators, providing operators that are relevant for the classification of quasi exactly solvable, finite difference equations.
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.
We cast the $q$-rotor in the framework of Barnett-Pegg theory for rotation angle, whose underlying algebra is $SU_q(2)$. A new method to fix the deformation parameter from the theory is suggested. We test our ideas by fitting rotational spectra in deformed even-even and superdeformed nuclei. The results are in good agreement with the previous phenomenological applications of $q$-rotor model.
We examine a two-parameter ($\hbar ,$ $λ$) deformation of the Poincarè algebra which is covariant under the action of $SL_q(2,C).$ When $λ\rightarrow 0$ it yields the Poincarè algebra, while in the $\hbar\rightarrow 0$ limit we recover the classical quadratic algebra discussed previously in \cite{ssy95}, \cite{sy95}. The analogues of the Pauli-Lubanski vector $w$ and Casimirs $p^2$ and $w^2$ are found and a set of mutually commuting operators is constructed.
We study junctions of Wilson lines in refined SU(N) Chern-Simons theory and their local relations. We focus on junctions of Wilson lines in antisymmetric and symmetric powers of the fundamental representation and propose a set of local relations which realize one-parameter deformations of quantum groups $\dot{U}_{q}(\mathfrak{sl}_{m})$ and $\dot{U}_{q}(\mathfrak{sl}_{n|m})$.
