Related papers: Wigner Oscillators, Twisted Hopf Algebras and Seco…

A supersymmetric extension of the Hahn algebra is introduced. This quadratic superalgebra, which we call the Hahn superalgebra, is constructed using the realization provided by the Dunkl oscillator model in the plane, whose Hamiltonian…

Certain types of generalized undeformed and deformed boson algebras which admit a Hopf algebra structure are introduced, together with their Fock-type representations and their corresponding $R$-matrices. It is also shown that a class of…

This paper generalizes the Drinfel'd twist to the multiplier Hopf algebra case. For a multiplier Hopf algebra $A$ with a twist $J$, we construct a new multiplier Hopf algebra $A^{J}$. Furthermore, if $A$ is quasitriangular, then $A^{J}$ is…

We construct an universal enveloping algebra associated to the ternary extension of Lie (super)algebras called Lie algebra of order three. A Poincar\'e-Birkhoff-Witt theorem is proven is this context. It this then shown that this universal…

Twisted deformations of the conformal symmetry in the Hopf algebraic framework are constructed. The first one is obtained by a Jordanian twist built up from dilatation and momenta generators. The second is the light-like…

We deform Heisenberg algebra and corresponding coalgebra by twist. We present undeformed and deformed tensor identities. Coalgebras for the generalized Poincar\'{e} algebras have been constructed. The exact universal $R$-matrix for the…

The recent focus on deformations of algebras called quantum algebras can be attributed to the fact that they appear to be the basic algebraic structures underlying an amazingly diverse set of physical situations. To date many interesting…

For a Hopf algebra B with bijective antipode, we show that the Heisenberg double H(B^*) is a braided commutative Yetter--Drinfeld module algebra over the Drinfeld double D(B). The braiding structure allows generalizing H(B^*) =…

We give some applications of a Hopf algebra constructed from a group acting on another Hopf algebra A as Hopf automorphisms, namely Molnar's smash coproduct Hopf algebra. We find connections between the exponent and Frobenius-Schur…

We present an elementary construction of a (highly degenerate) Hopf pairing between the universal enveloping algebra $U(\mathfrak{g})$ of a finite-dimensional Lie algebra $\mathfrak{g}$ over arbitrary field $\mathbf{k}$ and the Hopf algebra…

By using cocycle deformation, we construct a certain class of Hopf algebras, containing the quantized enveloping algebras and their analogues, from what we call pre-Nichols algebras. Our construction generalizes in some sense the known…

Given a Hopf algebra $H$ and a counital $2$-cocycle $\mu$ on $H$, Drinfeld introduced a notion of twist which deforms an $H$-module algebra $A$ into a new algebra $A_\mu$. We show that when $A$ is a quadratic algebra, and $H$ acts on $A$ by…

We study the relationship between antipodes on a Hopf algebroid $\mathcal{H}$ in the sense of B\"ohm-Szlachanyi and the group of twists that lies inside the associated convolution algebra. We specialize to the case of a faithfully flat…

The quantum phase space described by Heisenberg algebra possesses undeformed Hopf algebroid structure. The $\kappa$-deformed phase space with noncommutative coordinates is realized in terms of undeformed quantum phase space. There are…

Non-commutative Quantum Mechanics in 3D is investigated in the framework of the abelian Drinfeld twist which deforms a given Hopf algebra while preserving its Hopf algebra structure. Composite operators (of coordinates and momenta) entering…

Noncommutative oscillators are first-quantized through an abelian Drinfel'd twist deformation of a Hopf algebra and its action on a module. Several important and subtle issues making possible the quantization are solved. The spectrum of the…

We extend the construction of 2D superintegrable Hamiltonians with separation of variables in spherical coordinates using combinations of shift, ladder, and supercharge operators to models involving rational extensions of the two-parameter…

We construct explicit Drinfel'd twists for the generalized Cartan type $H$ Lie algebras in characteristic $0$ and obtain the corresponding quantizations and their integral forms. Via making modular reductions including modulo $p$ reduction…

Given a Hopf algebra $H$ and a projection $H\to A$ to a Hopf subalgebra, we construct a Hopf algebra $r(H)$, called the partial dualization of $H$, with a projection to the Hopf algebra dual to $A$. This construction provides powerful…

We re-examine all the contractions related with the ${\cal U}_q(su(2))$ deformed algebra and study the consequences that the contraction process has for their structure. We also show using ${\cal U}_q(su(2))\times{\cal U}(u(1))$ as an…