Sultan A. Celik

Differential geometry of the quantum 3-dimensional space

We present a differential calculus on the extension of the quantum plane obtained considering that the (bosonic) generator x is invertible and furthermore working polynomials in ln x instead of polynomials in x. We call quantum Lie algebra to this extension and we obtain its Hopf algebra structure and its dual Hopf algebra.

On the differential geometry of

The differential calculus on the quantum supergroup GL

COMMENT ON THE DIFFERENTIAL CALCULUS ON THE QUANTUM EXTERIOR PLANE

_q(1| 1)

Differential calculus on the h-superplane

was introduced by Schmidke {\it et al}. (1990 {\it Z. Phys. C} {\bf 48} 249). We construct a differential calculus on the quantum supergroup GL

DIFFERENTIAL GEOMETRY OF THE q-PLANE

_q(1| 1)

Two-parametric extension of h-deformation of Gr(1|1)

in a different way and we obtain its quantum superalgebra. The main structures are derived without an R-matrix. It is seen that the found results can be written with help of a matrix

Two-parameter nonstandard deformation of 2×2 matrices

\hat{R}

TWO-PARAMETER DIFFERENTIAL CALCULUS ON THE h-EXTERIOR PLANE

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 GLp,q(2). We also give a two-parameter deformation of the fermionic oscillator algebra.

A Differential Calculus on the (h, j)-Deformed Z3-Graded Superplane

A noncommutative differential calculus on the h-superplane is presented via a contraction of the q-superplane. An R-matrix which satisfies both ungraded and graded Yang–Baxter equations is obtained and a new deformation of the (1+1)-dimensional classical phase space (the super-Heisenberg algebra) is introduced.