相关论文: On Framed Quantum Principal Bundles
A construction of the noncommutative-geometric counterparts of classical classifying spaces is presented, for general compact matrix quantum structure groups. A quantum analogue of the classical concept of the classifying map is introduced…
We review quantum gravity model building using the new formalism of `quantum Riemannian geometry' to construct this on finite discrete spaces and on fuzzy ones such as matrix algebras. The formalism starts with a `differential structure' as…
These notes are intended as an introduction to a study of applications of noncommutative calculus to quantum statistical Physics. Centered on noncommutative calculus we describe the physical concepts and mathematical structures appearing in…
Let $\Uq$ be a quantum group. Regarding a (noncommutative) space with $\Uq$-symmetry as a $\Uq$-module algebra $A$, we may think of equivariant vector bundles on $A$ as projective $A$-modules with compatible $\Uq$-action. We construct an…
This work concerns some issues about the interplay of standard and geometric (Hamiltonian) approaches to finite-dimensional quantum mechanics, formulated in the projective space. Our analysis relies upon the notion and the properties of…
Recent work suggests that topological features of certain quantum gravity theories can be interpreted as particles, matching the known fermions and bosons of the first generation in the Standard Model. This is achieved by identifying…
A non--commutative analogue of the classical differential forms is constructed on the phase--space of an arbitrary quantum system. The non--commutative forms are universal and are related to the quantum mechanical dynamics in the same way…
Classification theory and the study of projective varieties which are covered by rational curves of minimal degrees naturally leads to the study of families of singular rational curves. Since families of arbitrarily singular curves are hard…
The past few years have seen a revived interest in quantum geometrical characterizations of band structures due to the rapid development of topological insulators and semi-metals. Although the metric tensor has been connected to many…
Following the approach of Budzy\'nski and Kondracki, we define covariant differential algebras and connections on locally trivial quantum principal fibre bundles. We also consider covariant derivatives, connection forms and curvatures and…
The use of geometric invariants has recently played an important role in the solution of classification problems in non-commutative ring theory. We construct geometric invariants of non-commutative projectivizations, a significant class of…
We investigate the quantum spectrum and Gamma structure for projective bundles, blow-ups, and standard flips. After restricting the quantum multiplication to the exceptional curve direction, we obtain a decomposition of the quantum…
The leitmotiv of this review is noncommutative principal U(1)-bundles and associated line bundles. In the first part I give a brief introduction to Hopf-Galois theory and its applications, from field extensions to principal group actions. I…
We introduce a convenient framework for constructing and analyzing orthogonal Thom spectra arising from virtual vector bundles. This framework enables us to set up a theory of orientations and graded Thom isomorphisms with good…
The central structure in various versions of noncommutative geometry is a differential calculus on an associative algebra. This is an analogue of the calculus of differential forms on a manifold. In this short review we collect examples of…
Quantum groups have been widely explored as a tool to encode possible nontrivial generalisations of reference frame transformations, relevant in quantum gravity. In quantum information, it was found that the reference frames can be…
This is a self-contained introduction to quantum Riemannian geometry based on quantum groups as frame groups, and its proposed role in quantum gravity. Much of the article is about the generalisation of classical Riemannian geometry that…
The conditions under which a given manifold $M$ may be given a tangent bundle or a cotangent bundle structure are analyzed. This is an important property arising in different contexts. For instance, in the study of integrability of a given…
In this survey article we give basic introduction to the theory of quantum families of maps. We begin with a general look at non-commutative (or "quantum") topology. Then we formulate all our results in this language. Existence of quantum…
The purpose of this contribution is to give an introduction to quantum geometry and loop quantum gravity for a wide audience of both physicists and mathematicians. From a physical point of view the emphasis will be on conceptual issues…