Quantum double of Heisenberg-Weyl algebra, its universal R-matrix and their representations

Hopf algebra quantizations of 4-dimensional and 6-dimensional real classical Drinfel'd doubles are studied by following a direct "analytic" approach. The full quantization is explicitly obtained for most of the Drinfel'd doubles, except a small number of them for which the dual Lie algebra is either sl(2) or so(3). In the latter cases, the classical r-matrices underlying the Drinfel'd double quantizations contain known standard ones plus additional twists. Several new four an...

Yangian Double $DY(A(m,n))$ of Lie Superalgebra $A(m,n)$ is described in terms of generators and defining relations. It is proved triangular decomposition for Yangian $Y(A(m,n))$ and its quantum double $DY(A(m,n))$ as a corollary of PBW theorem. It is introduced the normally ordered bases in Yangian and its dual Hopf superalgebra in quantum double. It is calculated the pairing formulas between the elements of its bases. It is received the formula of Universal $R$-matrix of Ya...

We give a construction of Drienfeld's quantum double for a nonstandard deformation of Borel subalgebra of $sl(2)$. We construct explicitly some simple representations of this quantum algebra and from the universal R-matrix we obtain the explicit solutions of the Yang-Baxter equation in those cases.

This paper is an extended version of our previous short letter \cite{ZG2} and is attempted to give a detailed account for the results presented in that paper. Let $U_q({\cal G}^{(1)})$ be the quantized nontwisted affine Lie algebra and $U_q({\cal G})$ be the corresponding quantum simple Lie algebra. Using the previous obtained universal $R$-matrix for $U_q(A_1^{(1)})$ and $U_q(A_2^{(1)})$, we determine the explicitly spectral-dependent universal $R$-matrix for $U_q(A_1)$ and ...

The Heisenberg double $D_q(E_2)$ of the quantum Euclidean group $\mathcal{O}_q(E_2)$ is the smash product of $\mathcal{O}_q(E_2)$ with its Hopf dual $U_q(\mathfrak{e}_2)$. For the algebra $D_q(E_2)$, explicit descriptions of its prime, primitive, and maximal spectra are obtained. All prime factors of $D_q(E_2)$ are presented as generalized Weyl algebras. As a result, we obtain that the algebra $D_q(E_2)$ has no finite-dimensional representations, and that $D_q(E_2)$ cannot ha...

A $Z_3$-graded Hopf algebra structure of exterior algebra with two generators is introduced. Two covariant differential calculus on the $Z_3$-graded exterior algebra are presented. Using the generators and their partial derivatives a Grassmann-Heisenberg algebra is constructed. An R-matrix which satisfies graded Yang-Baxter equations is obtained. A $Z_3$-graded universal enveloping algebra $U_q(\widetilde{gl}(2))$ is constructed with the quadratic elements of the Grassmann-We...

Using the previous obtained universal $R$-matrix for the quantized nontwisted affine Lie algebras $U_q(A_1^{(1)})$ and $U_q(A_2^{(1)})$, we determine the explicitly spectral-dependent universal $R$-matrix for the corresponding quantum Lie algebras $U_q(A_1)$ and $U_q(A_2)$. As their applications, we reproduce the well-known results in the fundamental representations and we also derive an extreamly explicit formula of the spectral-dependent $R$-matrix for the adjoint represent...

In order to assess possible observable effects of noncommutativity in deformations of quantum mechanics, all irreducible representations of the noncommutative Heisenberg algebra and Weyl-Heisenberg group on the two-torus are constructed. This analysis extends the well known situation for the noncommutative torus based on the algebra of the noncommuting position operators only. When considering the dynamics of a free particle for any of the identified representations, no obser...

We write the fermionic $q$-Fock space representation of $U_q(\hat{sl_n})$ as an infinite extended braided tensor product of finite-dimensional fermionic $U_q(sl_n)$-quantum planes or exterior algebras. Using braided geometrical techniques developed for such quantum exterior algebras, we provide a new approach to the Kashiwara-Miwa-Stern action of the Heisenberg algebra on the $q$-fermionic Fock space, obtaining the action in detail for the lowest nontrivial case $[b_{2},b_{-2...

Twisted tensor powers of quasitriangular Hopf algebras with diagonal sub-Hopf-algebras (self-diagonal tensor powers) are introduced together with their duals and their mutual *-structures as generalizations of the Drinfel'd double as given by Reshetikhin and Semenov-Tian-Shansky. R-Matrices for tensor products of representations are derived.