Related papers: The quantum Teichmuller space as a noncommutative …
In this paper, we study the asymptotic geometry of Teichmuller space of Riemann surfaces and give bounds on the Weil-Petersson sectional curvature of Teichmuller space, in terms of the length of the shortest geodesic on the surface. This…
This is an introduction to an algebraic construction of a gravity theory on noncommutative spaces which is based on a deformed algebra of (infinitesimal) diffeomorphisms. We start with some fundamental ideas and concepts of noncommutative…
In this chapter, we survey the algebraic aspects of quantum Teichm\"uller space, generalized Kashaev algebra and a natural relationship between the two algebras.
We show that the reduced quantum hyperbolic invariants of pseudo-Anosov diffeomorphisms of punctured surfaces are intertwiners of local representations of the quantum Teichm\"uller spaces. We characterize them as the only intertwiners that…
Based on hyperbolic geometric considerations, Roger and Yang introduced an extension of the Kauffman bracket skein algebra that includes arcs. In particular, their skein algebra is a deformation quantization of a certain commutative curve…
We prove that the quantum moduli algebra associated to a possibly punctured compact oriented surface and a complex semisimple Lie algebra $\mathfrak{g}$ is a Noetherian and finitely generated ring. If the surface has punctures, we prove…
A covariant functor from the category of generic complex algebraic curves to a category of the AF-algebras is constructed. The construction is based on a representation of the Teichmueller space of a curve by the measured foliations due to…
Our experience shows that dealing with noncommutative objects one should not imitate the classical commutative mathematics, but follow "the way it is" starting with basics. In this paper we consider mainly two such problems: noncommutative…
In analogy with conventional quantum mechanics, non-commutative quantum mechanics is formulated as a quantum system on the Hilbert space of Hilbert-Schmidt operators acting on non-commutative configuration space. It is argued that the…
It is clarified that Heisenberg quantization was proposed in empty space. Based on established experiments, the generalized Heisenberg quantization in physical space is obtained. Physical space quantization includes important new physics:…
A quantum deformation of 3-dimensional lattice gauge theory is defined by applying the Reshetikhin-Turaev functor to a Heegaard diagram associated to a given cell complex. In the root-of-unity case, the construction is carried out with a…
We discuss examples of non-commutative spaces over non-archimedean fields. Those include non-commutative and quantum affinoid algebras, quantized K3 surfaces and quantized locally analytic p-adic groups.
The quantum Heisenberg manifolds are noncommutive manifolds constructed by M. Rieffel as strict deformation quantizations of Heisenberg manifolds and have been studied by various authors. Rieffel constructed the quantum Heisenberg manifolds…
We demonstrate that dynamical noncommutative space-time will give rise to deformed oscillator algebras. In turn, starting from some q-deformations of these algebras in a two dimensional space for which the entire deformed Fock space can be…
The problem of holomorphic contractibilty of the Teichmuller spaces $T(0, n)$ of punctured spheres ($n > 4$) arose in the 1970s in connection with solving algebraic equations in Banach algebras. Recently it was solved by the author in…
Topological complexity for spaces was introduced by M. Farber as a minimal number of continuity domains for motion planning algorithms. It turns out that this notion can be extended to the case of not necessarily commutative C*-algebras.…
Following ideas from a preprint of the second author, see [2], we investigate relations of dynamical Teichmuller spaces with dynamical objects. We also establish some connections with the theory of deformations of inverse limits and…
Quantization of the Teichm\"uller space of a non-compact Riemann surface has emerged in 1980's as an approach to three dimensional quantum gravity. For any choice of an ideal triangulation of the surface, Thurston's shear coordinate…
We introduce the notion of a quantum locally compact metric space, which is the noncommutative analogue of a locally compact metric space, and generalize to the nonunital setting the notion of quantum metric spaces introduced by Rieffel. We…
These are expanded lecture notes of a mini-course whose objectives were to introduce the basic concepts, constructions and techniques of noncommutative geometry, as well as their uses as a framework for modelling quantum spacetime. Key…