Related papers: Dunkl Operators as Covariant Derivatives in a Quan…
We give a construction of a Dirac operator on a quantum group based on any simple Lie algebra of classical type. The Dirac operator is an element in the vector space $U_q(\g) \otimes \mathrm{cl}_q(\g)$ where the second tensor factor is a…
We study conformal invariants that arise from functions in the nullspace of conformally covariant differential operators. The invariants include nodal sets and the topology of nodal domains of eigenfunctions in the kernel of GJMS operators.…
Symmetries are crucial for tailoring parametrized quantum circuits to applications, due to their capability to capture the essence of physical systems. In this work, we shift the focus away from incorporating symmetries in the circuit…
Symmetry is present in many tasks in computer vision, where the same class of objects can appear transformed, e.g. rotated due to different camera orientations, or scaled due to perspective. The knowledge of such symmetries in data coupled…
Dunkl operators are differential-difference operators parametrized by a finite reflection group and a weight function. The commutative algebra generated by these operators generalizes the algebra of standard differential operators and…
We develop the theory of quantum (a.k.a. noncommutative) relations and quantum (a.k.a. noncommutative) graphs in the finite-dimensional covariant setting, where all systems (finite-dimensional $C^*$-algebras) carry an action of a compact…
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…
The work analyzes the theory of Dunkl operator as a moment differential operator. This last operator generalizes the first one whenever the sequence of moments satisfies appropriate classical properties, classically considered in the…
A noncommutative-geometric generalization of classical Weil theory of characteristic classes is presented, in the conceptual framework of quantum principal bundles. A particular care is given to the case when the bundle does not admit…
It is a classic result that the geometry of the total space of a principal bundle with reference to the action of the bundle's structure group is codified in the bundle's operation, a collection of derivations comprising the de Rham…
Let M be a simply connected Riemannian symmetric space, with at most one flat direction. We show that every Riemannian (or unitary) vector bundle with parallel curvature over M is an associated vector bundle of a canonical principal bundle,…
Continuing our study of spectral triples on quantum domains, we look at unbounded invariant and covariant derivations in the quantum annulus. In particular, we investigate whether such derivations can be implemented by operators with…
Dynamical evolution is described as a parallel section on an infinite dimensional Hilbert bundle over the base manifold of all frames of reference. The parallel section is defined by an operator-valued connection whose components are the…
We present weighted covariant derivatives and wave operators for perturbations of certain algebraically special Einstein spacetimes in arbitrary dimensions, under which the Teukolsky and related equations become weighted wave equations. We…
The non-commutative differential calculus on quantum groups can be extended by introducing, in analogy with the classical case, inner product operators and Lie derivatives. For the case of $\GL$ we show how this extended calculus induces by…
Using the higher covariant derivative on a manifold $ M $ equipped with a torsion-free connection, we define a natural surjective bundle map $ \Phi $ from $ (\otimes(TM))\otimes (\wedge(TM)) $ to the vector bundle $ \mathcal{U}(M) $ of de…
The formulation of Geometric Quantization contains several axioms and assumptions. We show that for real polarizations we can generalize the standard geometric quantization procedure by introducing an arbitrary connection on the…
We investigate the interaction between the existence of reproducing kernels on infinite-dimensional Hermitian vector bundles and the positivity properties of the corresponding bundles. The positivity refers to the curvature form of certain…
A new functional calculus, developed recently for a fully non-perturbative treatment of quantum gravity, is used to begin a systematic construction of a quantum theory of geometry. Regulated operators corresponding to areas of 2-surfaces…
The geometry of the total space of a principal bundle with regard to the action of the bundle's structure group is elegantly described by the bundle's operation, a collection of derivations consisting of the de Rham differential and the…