Related papers: Dunkl Operators as Covariant Derivatives in a Quan…

The main features of homogeneous Cowen-Douglas operators, well-known for the unit disk, are generalized to the setting of hermitian bounded symmetric domains of arbitrary rank.

A formulation of quantum mechanics with additive and multiplicative (q-)difference operators instead of differential operators is studied from first principles. Borel-quantisation on smooth configuration spaces is used as guiding…

We give a direct calculation of the curvature of the Hitchin connection, in geometric quantization on a symplectic manifold, using only differential geometric techniques. In particular, we establish that the curvature acts as a first-order…

Applications to quantum gravity of some results in C*-algebras are developed. We open by describing why algebra may be an integral aspect of quantum gravity. By interpreting the inner automorphisms of a C*-algebra as families of parallel…

A key problem in the attempt to quantize the gravitational field is the choice of boundary conditions. These are mixed, in that spatial and normal components of metric perturbations obey different sets of boundary conditions. In the…

Discrete vector bundles are important in Physics and recently found remarkable applications in Computer Graphics. This article approaches discrete bundles from the viewpoint of Discrete Differential Geometry, including a complete…

We propose a new point of view on quantum cohomology, strongly motivated by the work of Givental and Dubrovin, but closer to differential geometry than the existing approaches. The central object is the D-module which "quantizes" a…

We conjecture a geometrical form of the Paley-Wiener theorem for the Dunkl transform and prove three instances thereof, one of which involves a limit transition from Opdam's results for the graded Hecke algebra. Furthermore, the connection…

We develop the first steps towards an analysis of geometry on the quantum spacetime proposed in [1]. The homogeneous elements of the universal differential algebra are naturally identified with operators living in tensor powers of Quantum…

We obtain spectral asymptotics for the quantized derivatives of elements from the first-order homogeneous Sobolev space on the quantum Euclidean space, extending an earlier result of McDonald, Sukochev and Xiong (Commun. Math. Phys. 2020).…

Noncommutative K\"ahler structures were recently introduced as an algebraic framework for studying noncommutative complex geometry on quantum homogeneous spaces. In this paper, we introduce the notion of a \emph{compact quantum homogeneous…

We outline the recent classification of differential structures for all main classes of quantum groups. We also outline the algebraic notion of `quantum manifold' and `quantum Riemannian manifold' based on quantum group principal bundles, a…

We semiclassicalise the theory of quantum group principal bundles to the level of Poisson geometry. The total space $X$ is a Poisson manifold with Poisson-compatible contravariant connection, the fibre is a Poisson-Lie group in the sense of…

Let X be a smooth projective curve over a field k of characteristic zero. The differential fundamental group of X is defined as the Tannakian dual to the category of vector bundles with (integrable) connections on X. This work investigates…

In recent years, algebras and modules of differential operators have been extensively studied. Equivariant quantization and dequantization establish a tight link between invariant operators connecting modules of differential operators on…

We determine linear inequalities on the eigenvalues of curvature operators that imply vanishing of the twisted $\hat A$ genus on a closed Riemannian spin manifold, where the twisting bundle is any prescribed parallel bundle of tensors.…

The characteristic forms in the bundle of connections of a principal bundle P over M determine the characteristic classes of P for degree less or equal to the dimension of M, and differential forms on the space of connections for higher…

We describe the time evolution of quantum systems in a classical background space-time by means of a covariant derivative in an infinite dimensional vector bundle. The corresponding parallel transport operator along a timelike curve $\cC$…

Let $E_1,\dots ,E_k$ and $E$ be natural vector bundles defined over the category $\Cal Mf_m^+$ of smooth oriented $m$--dimensional manifolds and orientation preserving local diffeomorphisms, with $m\geq 2$. Let $M$ be an object of $\Cal…

It is shown that every quantum principal bundle is braided, in the sense that there exists an intrinsic braid operator twisting the functions on the bundle. A detailed algebraic analysis of this operator is performed. In particular, it…