Related papers: Frame formalism for the N-dimensional quantum Eucl…
We briefly report our application of a version of noncommutative geometry to the quantum Euclidean space $R^N_q$, for any $N \ge 3$; this space is covariant under the action of the quantum group $SO_q(N)$, and two covariant differential…
We apply one of the formalisms of noncommutative geometry to $R^N_q$, the quantum space covariant under the quantum group $SO_q(N)$. Over $R^N_q$ there are two $SO_q(N)$-covariant differential calculi. For each we find a frame, a metric and…
A detailed study is made of the noncommutative geometry of $R^3_q$, the quantum space covariant under the quantum group $SO_q(3)$. For each of its two $SO_q(3)$-covariant differential calculi we find its metric, the corresponding frame and…
We briefly describe how to introduce the basic notions of noncommutative differential geometry on the 3-dim quantum space covariant under the quantum group of rotations $SO_q(3)$.
We briefly describe our application of a version of noncommutative differential geometry to the 3-dim quantum space covariant under the quantum group of rotations $SO_q(3)$ and sketch how this might be used to determine the correct physical…
The algebraic formulation of the quantum group gauge models in the framework of the $R$-matrix approach to the theory of quantum groups is given. We consider gauge groups taking values in the quantum groups and noncommutative gauge fields…
We report on our recent breakthrough in the costructionfor q>0 of Hermitean and "tractable" differential operators out of the U_qso(N)-covariant differential calculus on the noncommutative manifolds R_q^N (the socalled "quantum Euclidean…
The algebraic formulation of the quantum group covariant noncommutative geometry in the framework of the $R$-matrix approach to the theory of quantum groups is given. We consider structure groups taking values in the quantum groups and…
We show that the isotropic harmonic oscillator in the ordinary euclidean space ${\bf R}^N$ ($N\ge 3$) admits a natural q-deformation into a new quantum mechanical model having a q-deformed symmetry (in the sense of quantum groups),…
Invariant integrals of functions and forms over $q$ - deformed Euclidean space and spheres in $N$ dimensions are defined and shown to be positive definite, compatible with the star - structure and to satisfy a cyclic property involving the…
The quantum Euclidean space R_{q}^{N} is a kind of noncommutative space which is obtained from ordinary Euclidean space R^{N} by deformation with parameter q. When N is odd, the structure of this space is similar to R_{q}^{3}. Motivated by…
Different approaches are compared to formulation of quantum mechanics of a particle on the curved spaces. At first, the canonical, quasi-classical and path integration formalisms are considered for quantization of geodesic motion on the…
We develop a quantum effective action for scalar-tensor theories of gravity which is both spacetime diffeomorphism invariant and field reparameterisation (frame) invariant beyond the classical approximation. We achieve this by extending the…
This thesis is devoted to algorithmic aspects of the implementation of Cartan's moving frame method to the problem of the equivalence of submanifolds under a Lie group action. We adopt a general definition of a moving frame as an…
Quantum superalgebras $su_{q}(m\mid n)$ are studied in the framework of $R$-matrix formalism. Explicit parametrization of $L^{(+)}$ and $L^{(-)}$ matrices in terms of $su_{q}(m\mid n)$ generators are presented. We also show that quantum…
The measurement processes that are traditionally described within the realm of non-relativistic quantum mechanics are transcribed into the covariant framework of Cartan's space, the four-valued representation space of the restricted…
The quantum Euclidean spheres, $S_q^{N-1}$, are (noncommutative) homogeneous spaces of quantum orthogonal groups, $\SO_q(N)$. The *-algebra $A(S^{N-1}_q)$ of polynomial functions on each of these is given by generators and relations which…
The three-dimensional quantum Euclidean space is an example of a non-commutative space that is obtained from Euclidean space by $q$-deformation. Simultaneously, angular momentum is deformed to $so_q(3)$, it acts on the $q$-Euclidean space…
We show that the complicated *-structure characterizing for positive q the U_qso(N)-covariant differential calculus on the non-commutative manifold R_q^N boils down to similarity transformations involving the ribbon element of a central…
An embedding method to get $q$-deformations for the non--semisimple algebras generating the motion groups of $N$--dimensional flat spaces is presented. This method gives a global and simultaneous scheme of $q$-deformation for all $iso(p,q)$…