Related papers: A Differential Calculus on the $(h,j)$-Deformed Z$…
Based on results for real deformation parameter q we introduce a compact non- commutative structure covariant under the quantum group SOq(3) for q being a root of unity. To match the algebra of the q-deformed operators with necesarry…
We study the deformed tensor products of complex Hadamard matrices, $L_{ia,jb}=Q_{ib}H_{ij}K_{ab}$. One problem is that of reconstructing the quantum group $G_L\subset S_{NM}^+$ out of the quantum groups $G_H\subset S_N^+,G_K\subset S_M^+$…
We introduce a class of right $H$--covariant first--order differential calculi on principal comodule algebras generated by the Durdevi\'c braiding $\sigma$ and a chosen vertical ideal. Starting from the universal calculus, a strong…
Modular and quasimodular solutions of specific second order differential equation in the upper-half plane which originates from a study of supersingular j-invariants are given explicitly. A characterization of the differential equation is…
We study *-differential calculi over compact quantum groups in the sense of S.L. Woronowicz. Our principal results are the construction of a Hodge operator commuting with the Laplacian, the derivation of a corresponding Hodge decomposition…
Quantum deformations of sets of points of the real and the complexified projective line are constructed. These deformations depend on the deformation parameter q and certain further parameters \lambda_{ij}. The deformations for which the…
A *-product compatible with the comultiplication of the Hopf algebra of the functions on the Heisenberg group is determined by deforming a coboundary Lie-Poisson structure defined by a classical r-matrix satisfying the modified Yang-Baxter…
For transcendental values of $q$ all bicovariant first order differential calculi on the coordinate Hopf algebras of the quantum groups $SL_q(n+1)$ and $Sp_q(2n)$ are classified. It is shown that the irreducible bicovariant first order…
In this note we highlight a common origin for many ubiquitous geometric structures, as well as several new ones by using only the functors of differential calculus in A.M Vinogradov's original sense, adapted to special classes of (graded)…
The non-standard (Jordanian) quantum deformations of $so(2,2)$ and (2+1) Poincar\'e algebras are constructed by starting from a quantum $sl(2,\R)$ basis such that simple factorized expressions for their corresponding universal $R$-matrices…
We define a new ${\mathbb Z}_2$-graded quantum (2+1)-space and show that the extended ${\mathbb Z}_2$-graded algebra of polynomials on this ${\mathbb Z}_2$-graded quantum space, denoted by ${\cal F}({\mathbb C}_q^{2\vert1})$, is a ${\mathbb…
This manuscript introduces $J_3$-numbers, a seemingly missing three-dimensional intermediate between complex numbers related to points in the Cartesian coordinate plane and Hamilton's quaternions in the 4D space. The current development is…
Admissible structure constants related to the dual Lie superalgebras of particular Lie superalgebra $({\cal C}^3 + {\cal A})$ are found by straightforward calculations from the matrix form of super Jacobi and mixed super Jacobi identities…
We introduce and study the Koszul complex for a Hecke $R$-matrix. Its cohomologies, called the Berezinian, are used to define quantum superdeterminant for a Hecke $R$-matrix. Their behaviour with respect to Hecke sum of $R$-matrices is…
We first give a pedagogical introduction to the differential calculus on q-groups and analize the relation between differential calculus and q-Lie algebra. Equivalent definitions of bicovariant differential calculus are studied and their…
A triangular quantum deformation of $ osp(2/1) $ from the classical $r$-matrix including an odd generator is presented with its full Hopf algebra structure. The deformation maps, twisting element and tensor operators are considered for the…
We construct an associative differential algebra on a two-parameter quantum plane associated with a nilpotent endomorphism $d$ in the two cases $d^{2}=0$ and $d^3=0$ $(d^2\neq 0).$ The correspondent curvature is derived and the related non…
The $q$-calculus for generic $q$ is developed and related to the deformed oscillator of parameter $q^{1/2}$. By passing with care to the limit in which $q$ is a root of unity, one uncovers the full algebraic structure of ${{\cal…
Explicit general constructions of paragrassmann calculus with one and many variables are given. Relations of the paragrassmann calculus to quantum groups are outlined and possible physics applications are briefly discussed. This paper is…
A rigid framework for the Cartan calculus of Lie derivatives, inner derivations, functions, and forms is proposed. The construction employs a semi-direct product of two graded Hopf algebras, the respective super-extensions of the deformed…