We give a two-parameter quantum deformation of the exterior plane and its differential calculus without the use of any R-matrix and relate it to the differential calculus with the R-matrix. We prove that there are two types of solutions of the Yang–Baxter equation whose symmetry group is GL p,q(2). We also give a two-parameter deformation of the fermionic oscillator algebra.
We study differential calculus on h-deformed bosonic and fermionic quantum space. It is shown that the fermionic quantum space involves a parafermionic variable as well as a classical fermionic one. Further we construct the classical su(2) algebra on the fermionic quantum space and discuss a mapping between the classical su(2) and the h-deformed su(2) algebras.
In this paper, a generalization and extension of some recent results is proposed. Namely, a four-dimensional quantum space is constructed and a differential calculus, obeying the usual criteria of covariance, is built upon this space. These results are then used to construct a gauge theory on this four-dimensional quantum space.
We study quantum deformed gl(n) and igl(n) algebras on a quantum space discussing multi-parametric extension. We realize elements of deformed gl(n) and igl(n) algebras by a quantum fermionic space. We investigate a map between deformed igl(2) algebras of our basis and other basis.
In the paper a theoretical study the both the quantized energies of excitonic states and their wave functions in grapheneand in materials with "Mexican hat" band structure dispersion as well as in zinc-blende GaN is presented. An integral twodimensionalSchrödinger equation of the electron-hole pairing for a particles with electron-hole symmetry of reflection isexactly solved. The solutions of Schrödinger equation in momentum space in studied materials by projection the twodimensionalspace of momentum on the three-dimensional sphere are found exactly. We analytically solve an integral twodimensionalSchrödinger equation of the electron-hole pairing for particles with electron-hole symmetry of reflection. Instudied materials the electron-hole pairing leads to the exciton insulator states. Quantized spectral series and lightabsorption rates of the excitonic states which distribute in valence cone are found exactly. If the electron and hole areseparated, their energy is higher than if they are paired. The particle-hole symmetry of Dirac equation of layered materialsallows perfect pairing between electron Fermi sphere and hole Fermi sphere in the valence cone and conduction cone andhence driving the Cooper instability. The solutions of Coulomb problem of electron-hole pair does not depend from a widthof band gap of graphene. It means the absolute compliance with the cyclic geometry of diagrams at justification of theequation of motion for a microscopic dipole of graphene where >1 s r . The absorption spectrums for the zinc-blendeGaN/(Al,Ga)N quantum well as well as for the zinc-blende bulk GaN are presented. Comparison with availableexperimental data shows good agreement.
Fock space representation of differential calculus on the noncommutative quantum space