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Z$_3$-graded differential geometry of quantum plane  [PDF]
Salih Celik
Mathematics , 2002, DOI: 10.1088/0305-4470/35/30/308
Abstract: In this work, the Z$_3$-graded differential geometry of the quantum plane is constructed. The corresponding quantum Lie algebra and its Hopf algebra structure are obtained. The dual algebra, i.e. universal enveloping algebra of the quantum plane is explicitly constructed and an isomorphism between the quantum Lie algebra and the dual algebra is given.
Differential geometry of the q-quaternion  [PDF]
Salih Celik
Mathematics , 2001,
Abstract: Differential calculus on the quantum quaternionic group GL(1,H$_q$) is introduced.
Differential Geometry of the Lie algebra of the quantum plane  [PDF]
Salih ?elik,Sultan A. ?elik
Mathematics , 2003, DOI: 10.1007/s10582-006-0083-9
Abstract: 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. Differential geometry of the quantum Lie algebra of the extended quantum plane and its Hopf algebra structure is obtained. Its dual Hopf algebra is also given.
Differential geometry of the q-superplane  [PDF]
Salih Celik
Mathematics , 2001, DOI: 10.1088/0305-4470/31/48/006
Abstract: Hopf algebra structures on the extended q-superplane and its differential algebra are defined. An algebra of forms which are obtained from the generators of the extended q-superplane is introduced and its Hopf algebra structure is given
Bicovariant differential geometry of the quantum group GL_q(3)  [PDF]
Paolo Aschieri,Leonardo Castellani
Physics , 1992, DOI: 10.1016/0370-2693(92)90887-A
Abstract: We construct a bicovariant differential calculus on the quantum group $GL_q(3)$, and discuss its restriction to $[SU(3) \otimes U(1)]_q$. The $q$-algebra of Lie derivatives is found, as well as the Cartan-Maurer equations. All the quantities characterizing the non-commutative geometry of $GL_q(3)$ are given explicitly.
Differential geometry of GL$_{p,q}(1| 1)$  [PDF]
Salih Celik
Mathematics , 2001, DOI: 10.1063/1.1285032
Abstract: We introduce a construction of the differential calculus on the quantum supergroup GL$_{p,q}(1| 1)$. We obtain two differential calculi, respectively, associated with the left and right Cartan-Maurer one-forms. We also obtain the quantum superalgebra of GL$_{p,q}(1| 1)$. Although all of the structures we obtain are derived without an R matrix, they neverthless can be expressed using an R matrix.
On the Differential Geometry of $GL_q(1| 1)$  [PDF]
Salih Celik
Mathematics , 2001, DOI: 10.1088/0305-4470/31/48/005
Abstract: The differential calculus on the quantum supergroup GL$_q(1| 1)$ 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$_q(1| 1)$ 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 $\hat{R}$
Geometry of the reduced quantum plane  [PDF]
R. Coquereaux,A. O. Garcia,R. Trinchero
Physics , 1998,
Abstract: We consider the space M of NxN matrices as a reduced quantum plane and discuss its geometry under the action and coaction of finite dimensional quantum groups (a quotient of U_q(SL(2)), q being an N-th root of unity, and its dual). We also introduce a differential calculus for M: a quotient of the Wess Zumino complex. We shall restrict ourselves to the case N odd and often choose the particular value N=3. The present paper (to appear in the proceedings of the conference "Quantum Groups and Fundamental Physical Applications", Palerme, December 1997) is essentially a short version of math-ph/9807012.
Solutions for linear differential equations with meromorphic coefficients of (p,q)-order in the plane
Lei-Min Li,Ting-Bin Cao
Electronic Journal of Differential Equations , 2012,
Abstract: In this article we study the growth of meromorphic solutions of high order linear differential equations with meromorphic coefficients of (p,q)-order. We extend some previous results due to Belaidi, Cao-Xu-Chen, Kinnunen, Liu- Tu -Shi, and others.
Quantum Orthogonal Planes: ISO_{q,r}(N) and SO_{q,r}(N) -- Bicovariant Calculi and Differential Geometry on Quantum Minkowski Space  [PDF]
Paolo Aschieri,Leonardo Castellani,Antonio Maria Scarfone
Mathematics , 1997, DOI: 10.1007/s100529800968
Abstract: We construct differential calculi on multiparametric quantum orthogonal planes in any dimension N. These calculi are bicovariant under the action of the full inhomogeneous (multiparametric) quantum group ISO_{q,r}(N), and do contain dilatations. If we require bicovariance only under the quantum orthogonal group SO_{q,r}(N), the calculus on the q-plane can be expressed in terms of its coordinates x^a, differentials dx^a and partial derivatives \partial_a without the need of dilatations, thus generalizing known results to the multiparametric case. Using real forms that lead to the signature (n+1,m) with m = n-1, n, n+1 , we find ISO_{q,r}(n+1, m) and SO_{q,r}(n+1,m) bicovariant calculi on the multiparametric quantum spaces. The particular case of the quantum Minkowski space ISO_{q,r}(3,1)/SO_{q,r}(3,1) is treated in detail. The conjugated partial derivatives \partial_a* can be expressed as linear combinations of the \partial_a. This allows a deformation of the phase-space where no additional operators (besides x^a and p_a) are needed.
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