Related papers: Quantum Principal Bundles
Higher bundles are homotopy coherent generalisations of classical fibre bundles. They appear in numerous contexts in geometry, topology and physics. In particular, higher principal bundles provide the geometric framework for higher-group…
We show how quiver representations and their invariant theory natu- rally arise in the study of some moduli spaces parametrizing bundles dened on an algebraic curve, and how they lead to ne results regarding the geometry of these spaces.
We explore the differential geometry of finite sets where the differential structure is given by a quiver rather than as more usual by a graph. In the finite group case we show that the data for such a differential calculus is described by…
We discuss in some generality aspects of noncommutative differential geometry associated with reality conditions and with differential calculi. We then describe the differential calculus based on derivations as generalization of vector…
We give coordinate formula and geometric description of the curvature of the tensor product connection of linear connections on vector bundles with the same base manifold. We define the covariant differential of geometric fields of certain…
The main goal of this article is to construct some geometric invariants for the topology of the set $\mathcal{F}$ of flat connections on a principal $G$-bundle $P\,\longrightarrow\, M$. Although the characteristic classes of principal…
After recalling briefly some basic properties of the quantum group $GL_q(2)$, we study the quantum sphere $S_q^2$, quantum projective space $CP_q(N)$ and quantum Grassmannians as examples of complex (K\"{a}hler) quantum manifolds. The…
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…
Noncommutative geometry, in its many incarnations, appears at the crossroad of various researches in theoretical and mathematical physics: from models of quantum space-time (with or without breaking of Lorentz symmetry) to loop gravity and…
Caianiello's derivation of Quantum Geometry through an isometric embedding of the spacetime ({\bf M},\tilde{g}) in the pseudo-Riemannian structure ({\bf T^*M},g^*_{AB}) is reconsidered. In the new derivation, a non-linear connection and the…
Over the quantum weighted 1-dimensional complex projective spaces, called quantum teardrops, the quantum line bundles associated with the quantum principal U(1)-bundles introduced and studied by Brzezinski and Fairfax are explicitly…
We review the elementary theory of gauge fields and the Becchi-Rouet-Stora- Tyutin symmetry in the context of differential geometry. We emphasize the topological nature of this symmetry and discuss a double Chevalley-Eilenberg complex for…
This PHD thesis is concerned with uncertainty relations in quantum probability theory, state estimation in quantum stochastics, and natural bundles in differential geometry. After some comments on the nature and necessity of decoherence in…
Let $X$ be a differentiable manifold endowed with a transitive action $\alpha:A\times X\longrightarrow X$ of a Lie group $A$. Let $K$ be a Lie group. Under suitable technical assumptions, we give explicit classification theorems, in terms…
A class of differential calculi is explored which is determined by a set of automorphisms of the underlying associative algebra. Several examples are presented. In particular, differential calculi on the quantum plane, the $h$-deformed…
An algebraic framework for noncommutative bundles with (quantum) homogeneous fibres is proposed. The framework relies on the use of principal coalgebra extensions which play the role of principal bundles in noncommutative geometry which…
Quantum geometry is a differential geometry based on quantum mechanics. It is related to various transport and optical properties in condensed matter physics. The Zeeman quantum geometry is a generalization of quantum geometry including the…
Over the past five years, there has been significant progress on the problem of quantization of diffeomorphism covariant field theories with {\it local} degrees of freedom. The absence of a background space-time metric in these theories…
We address the recently introduced notions of generalized principal bundle and generalized principal connection by keeping track of global geometric properties through local coordinate transformation laws. This approach leads us to…
It is shown that every quantum principal bundle with a compact structure group is a Hopf-Galois extension. This property naturally extends to the level of general differential structures, so that every differential calculus over a quantum…