Related papers: Geometry of the reduced quantum plane
A generalization of the differential geometry of forms and vector fields to the case of quantum Lie algebras is given. In an abstract formulation that incorporates many existing examples of differential geometry on quantum groups, we…
The zero modes of the chiral SU(n) WZNW model give rise to an intertwining quantum matrix algebra A generated by an n x n matrix a=(a^i_\alpha) (with noncommuting entries) and by rational functions of n commuting elements q^{p_i}. We study…
Quantum computers hold great promise, but it remains a challenge to find efficient quantum circuits that solve interesting computational problems. We show that finding optimal quantum circuits is essentially equivalent to finding the…
There have been several propositions for a geometric and essentially non-linear formulation of quantum mechanics. From a purely mathematical point of view, the point of view of Jordan algebra theory might give new strength to such…
We study a multi-parametric family of quadratic algebras in four generators, which includes coordinate algebras of noncommutative four-planes and, as quotient algebras, noncommutative three spheres. Particular subfamilies comprise Sklyanin…
We give a path integral construction of the quantum mechanical partition function for gauged finite groups. Our construction gives the quantization of a system of $d$, $N\times N$ matrices invariant under the adjoint action of the symmetric…
Quantum algebras are a mathematical tool which provides us with a class of symmetries wider than that of Lie algebras, which are contained in the former as a special case. After a self-contained introduction to the necessary mathematical…
Divided into three parts, the first marks out enormous geometric issues with the notion of quasi-freenss of an algebra and seeks to replace this notion of formal smoothness with an approximation by means of a minimal unital commutative…
The sliced skein algebra of a closed surface of genus $g$ with $m$ punctures, $\mathfrak{S}=\Sigma_{g,m}$, is the quotient of the Kauffman bracket skein algebra $\mathcal{S}_\xi(\mathfrak{S})$ corresponding to fixing the scalar values of…
We study covariant differential calculus on the quantum spheres S_q^2N-1. Two classification results for covariant first order differential calculi are proved. As an important step towards a description of the noncommutative geometry of the…
In our previous publications we have introduced a differential calculus on the algebra $U(gl(m))$ based on a new form of the Leibniz rule which differs from that usually employed in Noncommutative Geometry. This differential calculus…
The kinematical setting of spherically symmetric quantum geometry, derived from the full theory of loop quantum gravity, is developed. This extends previous studies of homogeneous models to inhomogeneous ones where interesting field theory…
The main result of this paper is a construction of fundamental domains for certain group actions on Lorentz manifolds of constant curvature. We consider the simply connected Lie group G~, the universal cover of the group SU(1,1) of…
The superfield formulation of two - dimensional $N=4$ Extended Supersymmetric Quantum Mechanics (SQM) is described. It is shown that corresponding classical Lagrangian describes the motion in the conformally flat metric with additional…
In this paper, we systematically establish the mathematical foundation for the $\text{U}^N(1)$ quantum geometric tensor (QGT) of mixed states Explicitly, we present a description based on the $\text{U}^N(1)$ principal bundle and derive a…
A noncommutative-geometric generalization of the theory of principal bundles is sketched. A differential calculus over corresponding quantum principal bundles is analysed. The formalism of connections is presented. In particular, operators…
Suppose $S$ is an affine, noetherian scheme, $X$ is a separated, noetherian $S$-scheme, $\mathcal{E}$ is a coherent ${\mathcal{O}}_{X}$-bimodule and $\mathcal{I} \subset T(\mathcal{E})$ is a graded ideal. We study the geometry of the…
We describe geometrically the classical and quantum inhomogeneous groups $G_0=(SL(2, \BbbC)\triangleright \BbbC^2)$ and $G_1=(SL(2, \BbbC)\triangleright \BbbC^2)\triangleright \BbbC$ by studying explicitly their shape algebras as a spaces…
In Gen. Rel. Grav. (36, 111-126 (2004); in press, gr-qc/0410010) we have proposed a model unifying general relativity and quantum mechanics based on a noncommutative geometry. This geometry was developed in terms of a noncommutative algebra…
We give complete detail of the description of the GNS representation of the quantum plane $\cA$ and its dual $\hat{\cA}$ as a von-Neumann algebra. In particular we obtain a rather surprising result that the multiplicative unitary $W$ is…