Related papers: Deformed boson algebras and the quantum double con…
This is a report on recent progress concerning the interactions between derived algebraic geometry and deformation quantization. We present the notion of derived algebraic stacks, of shifted symplectic and Poisson structures, as well as the…
Quantum superintegrable systems in two dimensions are obtained from their classical counterparts, the quantum integrals of motion being obtained from the corresponding classical integrals by a symmetrization procedure. For each quantum…
We establish Drinfeld realization for the two-parameter twisted quantum affine algebras using a new method. The Hopf algebra structure for Drinfeld generators is given for both untwisted and twisted two-parameter quantum affine algebras,…
We construct quantum commutators on module-algebras of quasi-triangular Hopf algebras. These are quantum-group covariant, and have generalized antisymmetry and Leibniz properties. If the Hopf algebra is triangular they additionally satisfy…
In this paper, firstly, we use the bosonic oscillators to construct a two-parameter deformed Virasoro algebra, which is a non-multiplicative Hom-Lie algebra. Secondly, a non-trivial Hopf structure related to the two-parameter deformed…
Parabosonic algebra in infinite degrees of freedom is presented as a generalization of the bosonic algebra, from the viewpoints of both physics and mathematics. The notion of super-Hopf algebra is shortly discussed and the super-Hopf…
The concept of a quantum algebra is made easy through the investigation of the prototype algebras $u_{qp}(2)$, $su_q(2)$ and $u_{qp}(1,1)$. The latter quantum algebras are introduced as deformations of the corresponding Lie algebras~; this…
We present two classes of examples of Hopf algebroids associated with noncommutative principal bundles. The first comes from deforming the principal bundle while leaving unchanged the structure Hopf algebra. The second is related to…
Deformed gauge transformations on deformed coordinate spaces are considered for any Lie algebra. The representation theory of this gauge group forces us to work in a deformed Lie algebra as well. This deformation rests on a twisted Hopf…
An explicit construction of the braided dual of quantum $E(2)$ groups is described over the circle group $\mathbb{T}$ with respect to a specific $R$-matrix $R$. Additionally, the corresponding bosonization is also described.
We introduce a quasitriangular Hopf algebra or `quantum group' $U(B)$, the {\em double-bosonisation}, associated to every braided group $B$ in the category of $H$-modules over a quasitriangular Hopf algebra $H$, such that $B$ appears as the…
We find a dual version of a previous double-bosonisation theorem whereby each finite-dimensional braided-Hopf algebra $B$ in the category of comodules of a coquasitriangular Hopf algebra $A$ has an associated coquasitriangular Hopf algebra…
In recent papers of the author, a method was developed for constructing quasitriangular Hopf algebras (quantum groups) of the quantum-double type. As a by-product, a novel non-standard example of the quantum double has been found. In the…
We consider the conditions under which the $q$-oscillator algebra becomes a Hopf $*$-algebra. In particular, we show that there are at least two real forms associated with the algebra. Furthermore, through the representations, it is shown…
The deformed boson scheme in four kinds of boson operators, which was recently proposed by the present authors, is supplemented by the T-type deformation closely related with the su(1,1)-algebra. Two subjects are discussed in relation to…
In this paper as a continuation of Part I, the case of two kinds of boson operators is treated. The deformation of the coherent states for the su(2)- and the su(1,1)-algebra and their related deformed algebras are discussed in various forms…
We discuss how a q-mutation relation can be deformed replacing a pair of conjugate operators with two other and unrelated operators, as it is done in the construction of pseudo-fermions, pseudo-bosons and truncated pseudo-bosons. This…
We define and investigate pairings of multiplier Hopf algebras. It is shown that two dually paired regular multiplier Hopf ($*$-)algebras $A$ and $B$ yield a quantum double multiplier Hopf ($*$-)algebra which is again regular. Integrals on…
We introduce $*$-structures on braided groups and braided matrices. Using this, we show that the quantum double $D(U_q(su_2))$ can be viewed as the quantum algebra of observables of a quantum particle moving on a hyperboloid in q-Minkowski…
Contracting the $h$-deformation of $\SL(2,\Real)$, we construct a new deformation of two dimensional Poincar\'e algebra, the algebra of functions on its group and its differential structure. It is also shown that the Hopf algebra is…