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Let (\Gamma,d) be the 3D-calculus or the 4D_{\pm}-calculus on the quantum group SU_q(2). We describe all pairs (\pi, F) of a *-representation \pi of O(SU_q(2)) and of a symmetric operator F on the representation space satisfying a technical…
Diagonalization of a certain operator in irreducible representations of the positive discrete series of the quantum algebra U_q(su(1,1)) is studied. Spectrum and eigenfunctions of this operator are found in an explicit form. These…
The algebraic formulation of the quantum group covariant noncommutative geometry in the framework of the $R$-matrix approach to the theory of quantum groups is given. We consider structure groups taking values in the quantum groups and…
Hilbert space representations of cross product *-algebras of the Hopf *-algebras U_q(gl_2) with the coordinate algebras O(C^2_q) and O(R^3_q) of quantum vector spaces and U_q(su_2) with coordinate algebras O(SU_q(2)) and O(S^2_q) of…
Deformed orthogonal and pseudo-orthogonal Lie algebras are constructed which differ from deformations of Lie algebras in terms of Cartan subalgebra and root vectors and which make it possible to construct representations by operators acting…
We review the representation theory of the quantum group $U_\epsilon sl_2\mathbb{C}$ at a root of unity $\epsilon$ of odd order, focusing on geometric aspects related to the 3-dimensional quantum hyperbolic field theories (QHFT). Our…
In order to obtain a classification of all possible quantum deformations of the two-photon algebra $h_6$, we introduce its corresponding general Lie bialgebra, which is a coboundary one. Two non-standard quantum deformations of $h_6$,…
In this paper, we investigate finite-dimensional irreducible representations of the quantum affine general linear superalgebra $\mathrm{U}_q\big(\widehat{\mathfrak{gl}}_{m|n,\mathbf{s}}\big)$ for arbitrary 01-sequences $\mathbf{s}$, using…
In the paper we consider a realization of a finite dimensional irreducible representation of the Lie algebra $\mathfrak{gl}_n$ in the space of functions on the group $GL_n$. It is proved that functions corresponding to Gelfand-Tsetlin…
Generating functions for Clebsch-Gordan coefficients of osp(1|2) are derived. These coefficients are expressed as q goes to - 1 limits of the dual q-Hahn polynomials. The generating functions are obtained using two different approaches…
We consider two algebraic invariants in the representation theory of quantized enveloping algebras: the dimension growth of simple modules for the De Concini-Kac quantum group at roots of unity, and the Gelfand-Kirillov dimension of simple…
Quasisymmetric functions in superspace were introduced as a natural extension of classical quasisymmetric functions involving both commuting and anticommuting variables. In this paper, we first provide a characterization of the algebra of…
Some time ago, Rideau and Winternitz introduced a realization of the quantum algebra su_q(2) on a real two-dimensional sphere, or a real plane, and constructed a basis for its representations in terms of q-special functions, which can be…
In the theory of Clebsch-Gordan coefficients, one may recognize the domain space as the set of weakly semi-magic squares of size three. Two partitions on this set are considered: a triangle-hexagon model based on top lines, and one based on…
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…
An algebra homomorphism $\psi$ from the q-deformed algebra $U_q({\rm iso}_2)$ with generating elements $I$, $T_1$, $T_2$ and defining relations $[I,T_2]_q=T_1$, $[T_1,I]_q=T_2$, $[T_2,T_1]_q=0$ (where $[A,B]_q=q^{1/2}AB-q^{-1/2}BA$) to the…
Some introductory concepts and basic definitions of the Lie superalgebras and their quantum deformations are exposed. Especially the induced representation methods in both cases are described. Based on the Kac representation theory we have…
We study (quasi-)cohomological properties through an analysis of quantum Markov semi-groups. We construct higher order Hochschild cocycles using gradient forms associated with a quantum Markov semi-group. By using Schatten-$\mathcal{S}_p$…
After reviewing the recent results on the Drinfeld realization of the face type elliptic quantum group B_{q,lambda}(sl_N^) by the elliptic algebra U_{q,p}(sl_N^), we investigate a fusion of the vertex operators of U_{q,p}(sl_N^). The basic…
We obtain an interesting realization of the de Rham cohomological operators of differential geometry in terms of the noncommutative q-superoscillators for the supersymmetric quantum group GL_{qp} (1|1). In particular, we show that a unique…