Related papers: Representations of the q-deformed algebra $U_q({\r…
A $q$-analogue of the Higgs algebra, which describes the symmetry properties of the harmonic oscillator on the $2$-sphere, is obtained as the commutant of the $\mathfrak{o}_{q^{1/2}}(2) \oplus \mathfrak{o}_{q^{1/2}}(2)$ subalgebra of…
Representation theory of the quantum torus Hopf algebra, when the parameter $q$ is a root of unity, is studied. We investigate a decomposition map of the tensor product of two irreducibles into the direct sum of irreducibles, realized as a…
A class of highest weight irreducible representations of the algebra $U_h(A_\infty)$, the quantum analogue of the completion and central extension $A_\infty$ of the Lie algebra $gl_\infty$, is constructed. It is considerably larger than the…
Tensor products of irreducible representations of the Jordanian quantum algebras U_h(sl(2)) and U_h(su(1,1)) are considered. For both the highest weight finite dimensional representations of U_h(sl(2)) and lowest weight infinite dimensional…
Irreducible representations of quantum groups $SL_q(2)$ (in Woronowicz' approach) were classified in J.Wang, B.Parshall, Memoirs AMS 439 in the~case of $q$ being an~odd root of unity. Here we find the~irreducible representations for all…
A family of infinite-dimensional irreducible $*$-representations on $\mathcal{H}\simeq L^2(\mathbb{R})\otimes\mathbb{C}^N$ is defined for a quantum-deformed Lorentz algebra $\mathscr{U}_{\bf q}(sl_2)\otimes \mathscr{U}_{\widetilde{\bf…
We introduce a category of $q$-oscillator representations over the quantum affine superalgebras of type $D$ and construct a new family of its irreducible representations. Motivated by the theory of super duality, we show that these…
Let $\mathbb{F}$ be a field, and let $q\in\mathbb{F}$. The $q$-deformed Heisenberg algebra is the unital associative $\mathbb{F}$-algebra $\mathcal{H}(q)$ with generators $A,B$ and relation $AB-qBA=I$, where $I$ is the multiplicative…
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…
A universality of deformed Heisenberg algebra involving the reflection operator is revealed. It is shown that in addition to the well-known infinite-dimensional representations related to parabosons, the algebra has also finite-dimensional…
We calculate all irreducible representations over a subfamily of pointed Hopf algebras with group-likes the dihedral group analyzing the possible decompositions of the restriction to the dihedral group and calculating the Jacobson radical…
Using the second Drinfeld formulation of the quantized universal enveloping algebra $U_q(\widehat{sl_2})$ we introduce a family of its Heisenberg-type elements which are endowed with a deformed commutator and satisfy properties similar to…
In this paper, we study and classify Hilbert space representations of cross product *-algebras of the quantized enveloping algebra $U_q(e_2)$ with the coordinate algebras $O(E_q(2))$ of the quantum motion group and $O(\C_q)$ of the complex…
This lecture consists of two sections. In section 1 we consider the simplest version of a q-deformed Heisenberg algebra as an example of a noncommutative structure. We first derive a calculus entirely based on the algebra and then formulate…
A q-analogue of four dimensional conformally invariant field theory based on the quantum algebra U_{q}(so(4,2)) is proposed. The two- and three-point correlation functions are calculated. The construction is elaborated in order to fit the…
The $q$--deformation $U_q (h_4)$ of the harmonic oscillator algebra is defined and proved to be a Ribbon Hopf algebra.Associated with this Hopf algebra we define an infinite dimensional braid group representation on the Hilbert space of the…
2-Dim quantum Poincare` Group E_q(1,1) at roots of unity, its dual U_q(e(1,1)) and some of its homogeneous spaces are introduced. Invariant integrals on E_q(1,1) and its invariant discrete subgroup E(1,1\mid p) are constructed.…
This paper establishes several fundamental structural properties of the $q$-Heisenberg algebra $\mathfrak{h}_n(q)$, a quantum deformation of the classical Heisenberg algebra. We first prove that when $q$ is not a root of unity, the global…
Quantum groups at roots of unity have the property that their centre is enlarged. Polynomial equations relate the standard deformed Casimir operators and the new central elements. These relations are important from a physical point of view…
The nonabelian two-dimensional Lie algebra over a field $\mathbb{F}$ has a presentation by generators $A$, $B$ and relation $\left[ A,B\right]=A$, with the universal enveloping algebra having a presentation by generators $A$, $B$ and…