Related papers: Quantum Isometry Group for Spectral Triples with R…
Quantum groups emerged in the latter quarter of the 20th century as, on the one hand, a deep and natural generalisation of symmetry groups for certain integrable systems, and on the other as part of a generalisation of geometry itself…
We show that a certain geometric property, the QSF introduced by S. Brick and M. Mihalik, is universally true for {\ibf all} finitely presented groups $\Gamma$. One way of defining this property is the existence of a smooth compact manifold…
The notion of good spectral triple is initiated. We prove firstly that any regular spectral triple may be embedded in a good spectral triple, so that, in non-commutative geometry, we can restricts to deal only with good spectral triples.…
We consider some general aspects of the new noncommutative or quantum geometry coming out of the theory of quantum groups, in connection with Planck scale physics. A generalisation of Fourier or wave-particle duality on curved spaces…
The isospectral renormalization group is a powerful method to analyze the spectrum of operators in quantum field theory. It was introduced in 1995 [see \cite{BachFrohlichSigal1995}, \cite{BachFrohlichSigal1998}] and since then it has been…
Quantum relativity as a generalized, or rather deformed, version of Einstein relativity with a linear realization on a classical six-geometry beyond the familiar setting of space-time offer a new framework to think about the quantum…
The quantum field algebra of real scalar fields is shown to be an example of infinite dimensional quantum group. The underlying Hopf algebra is the symmetric algebra S(V) and the product is Wick's normal product. Two coquasitriangular…
We prove a number of results linking properties of actions by compact groups (both quantum and classical) on Banach spaces, such as uniform continuity, spectrum finiteness and extensibility of the actions across several constructions.…
We show that there exists a quantum compact metric space which underlies the setting of each Sobolev algebra associated to a subelliptic Laplacian $\Delta=-(X_1^2+\cdots+X_m^2)$ on a compact connected Lie group $G$ if $p$ is large enough,…
Quantum gravity was born as that branch of modern theoretical physics that tries to unify its guiding principles, i.e., quantum mechanics and general relativity. Nowadays it is providing new insight into the unification of all fundamental…
A notion of a quantum automorphism group of a finite quantum group, generalising that of a classical automorphism group of a finite group, is proposed and a corresponding existence result proved.
The notion of normal quantum subgroup introduced in algebraic context by Parshall and Wang when applied to compact quantum groups is shown to be equivalent to the notion of normal quantum subgroup introduced by the author. As applications,…
We present a general framework for automatic continuity results for groups of isometries of metric spaces. In particular, we prove automatic continuity property for the group of isometries of the Urysohn space and the Urysohn sphere, i.e.…
These notes summarise a talk surveying the combinatorial or Hamiltonian quantisation of three dimensional gravity in the Chern-Simons formulation, with an emphasis on the role of quantum groups and on the way the various physical constants…
In this paper we will show the following result: Let $\mathcal{N} $ be a complete (noncompact) connected orientable Riemannian three-manifold with nonnegative scalar curvature $S \geq 0$ and bounded sectional curvature $ K_{s} \leq K $.…
This is a self-contained review on the theory of quantum group and its applications to modern physics. A brief introduction is given to the Yang-Baxter equation in integrable quantum field theory and lattice statistical physics. The quantum…
A generalized Stiefel manifold is the manifold of orthonormal frames in a vector space with a non-degenerated bilinear or hermitian form. In this article, the Isometry group of the generalized Stiefel manifolds are computed at least up to…
Let v be the right regular representation of a compact quantum group G. Then (S.L.Woronowicz, "Compact quantum groups") v contains all irreducible representations of G and each irreducible representation enters v with the multiplicity equal…
Let X be an arbitrary hyperbolic geodesic metric space and let G be a countable non-elementary weakly acylindrical group of isometries of X. We show that the second bounded cohomology group of G with real coefficients or with coefficients…
The aim of the paper is to answer the following question: does $\kappa$-deformation fit into the framework of noncommutative geometry in the sense of spectral triples? Using a compactification of time, we get a discrete version of…