The Barbero-Immirzi parameter ($\gamma$) is introduced in loop quantum gravity (LQG) whose physical significance still a biggest open question; because of its profound traits. In some cases, it real-valued; while, complex-valued other cases. This emerges out the process denoting Lorentz connection with non compact group $SO(3,1)$ form complex values rotations, either $SO(3)$ or $SU(2)$. Initial...

The spectrum of eigenenergies of a quantum integrable system whose hamil-tonian depends on a single parameter shows degeneracies (crossings) when the parameter varies. We derive a semiclassical expression for the density of crossings in the plane energy-parameter, that is the number of crossings per unit of energy and unit of parameter, in terms of classical periodic orbits. We compare the resu...

we construct a family of two parameter quantum grou-ps‎ ‎$u_{r,s}(mathfrak{g})$ associated with a generalized kac-moody‎ ‎algebra corresponding to symmetrizable admissible borcherds cartan‎ ‎matrix‎. ‎we also construct the $textbf{a}$-form $u_{textbf{a}}$ and‎ ‎the classical limit of $u_{r,s}(mathfrak{g})$‎. ‎furthermore‎, ‎we‎ ‎display the equitable presentation for a subalgebra‎ ‎$u_{r,...

The spectrum of eigenenergies of a quantum integrable system whose hamiltonian depends on a single parameter shows degeneracies (crossings) when the parameter varies. We derive a semiclassical expression for the density of crossings in the plane energy-parameter, that is the number of crossings per unit of energy and unit of parameter, in terms of classical periodic orbits. We compare the resul...