Related papers: Quantum currents realization of the elliptic quant…
In a previous paper, we showed how one can obtain from the action of a locally compact quantum group on a type I-factor a possibly new locally compact quantum group. In another paper, we applied this construction method to the action of…
A Heisenberg-Clifford realization of a deformed $U(sl_{2})$ by two parameters $p$ and $q$ is discussed. The commutation relations for this deformed algebra have interesting connection with the theta functions.
Quantum computations that involve only Clifford operations are classically simulable despite the fact that they generate highly entangled states; this is the content of the Gottesman-Knill theorem. Here we isolate the ingredients of the…
The quantum dynamics of a particle in the Modified P\"oschl-Teller potential is derived from the group $SL(2,R)$ by applying a Group Approach to Quantization (GAQ). The explicit form of the Hamiltonian as well as the ladder operators is…
We investigate the structure of the elliptic algebra U_{q,p}(^sl_2) introduced earlier by one of the authors. Our construction is based on a new set of generating series in the quantum affine algebra U_q(^sl_2), which are elliptic analogs…
We give an integral representation for solutions of the elliptic quantum Knizhnik-Zamolodchikov-Bernard difference equations, in the case of sl(2). The result is based on a geometric construction of highest weight representations of the…
We define an elliptic version of the stable envelope of Maulik and Okounkov for the equivariant elliptic cohomology of cotangent bundles of Grassmannians. It is a version of the construction proposed by Aganagic and Okounkov and is based on…
This article surveys the application of the representation theory of loop groups to simple models in quantum field theory and to certain integrable systems. The common thread in the discussion is the construction of quantum fields using…
By encoding a qudit in a harmonic oscillator and investigating the infinite limit, we give an entirely new realization of continuous-variable quantum computation. The generalized Pauli group is generated by number and phase operators for…
The quantum group G_r,s provides a realisation of the two parameter quantum GL_p,q(2) which is known to be related to the two parameter nonstandard GL_hh'(2) group via a contraction method. We apply the contraction procedure to G_r,s and…
This note for the Proceedings of the International Congress of Mathematical Physics gives an account of a construction of an ``elliptic quantum group'' associated with each simple classical Lie algebra. It is closely related to elliptic…
We study quantum dynamical semigroups generated by noncommutative unbounded elliptic operators which can be written as Lindblad type unbounded generators. Under appropriate conditions, we first construct the minimal quantum dynamical…
We study properties of a scalar quantum field theory on the two-dimensional noncommutative plane with $E_q(2)$ quantum symmetry. We start from the consideration of a firstly quantized quantum particle on the noncommutative plane. Then we…
A manifestly Lorentz-covariant formulation of Loop Quantum Gravity (LQG) is given in terms of finite-dimensional representations of the Lorentz group. The formulation accounts for discrete symmetries, such as parity and time-reversal, and…
The paper contains the construction of a topological quantum field theory in which gluings along surfaces with boundary are allowed. The construction is made by using quantum deformations of sl(2,C). In the construction there appears a sign…
A realization of the elliptic quantum algebra $U_{q,p}(\widehat{sl_2})$ for any given level $k$ is constructed in terms of three free boson fields and their accompanying twisted partners. It can be viewed as the elliptic deformation of…
All quantum group structures on the group GL(2) are classified. It is shown that there are only two such structures, the well known quantum groups GL$_{qp}$(2) and GL$_{hh'}$(2).
The quantum field theories (QFT) constructed in [1,2] include phenomenology of interest. The constructions approximate: scattering by $1/r$ and Yukawa potentials in non-relativistic approximations; and the first contributing order of the…
To each representation of the elliptic quantum group $E_{\tau,\eta}(sl_2)$ is associated a family of commuting transfer matrices. We give common eigenvectors by a version of the algebraic Bethe ansatz method. Special cases of this…
We describe quantum theories for massless (p,q)-forms living on Kaehler spaces. In particular we consider four different types of quantum theories: two types involve gauge symmetries and two types are simpler theories without gauge…