相关论文: Quantum groups and q-lattices in phase space

We consider a twisted version of quantum groups corepresentations. This generalization amounts to include in the theory the case where quantum space coordinates and its endomorphism matrix entries belong to a non-commutative quadratic…

A new framework for noncommutative complex geometry on quantum homogeneous spaces is introduced. The main ingredients used are covariant differential calculi and Takeuchi's categorical equivalence for faithfully flat quantum homogeneous…

Duality transformations within the quantum mechanics of a finite number of degrees of freedom can be regarded as the dependence of the notion of a quantum, i.e., an elementary excitation of the vacuum, on the observer on classical phase…

We introduce the notion of noncommutative complex spheres with partial commutation relations for the coordinates. We compute the corresponding quantum symmetry groups of these spheres, and this yields new quantum unitary groups with partial…

The basic ideas in the theory of quantum mechanics on phase space are illustrated through an introduction of generalities, which seem to underlie most if not all such formulations and follow with examples taken primarily from kinematical…

We show the existence of a noncommutative spacetime structure in the context of a complete discussion on the underlying spacetime symmetries for the physical system of a free massless relativistic particle. The above spacetime symmetry…

We show that certain embeddable homogeneous spaces of a quantum group that do not correspond to a quantum subgroup still have the structure of quantum quotient spaces. We propose a construction of quantum fibre bundles on such spaces. 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…

This is an introduction to quantum algebra, from a geometric perspective. The classical spaces $X$, such as the Lie groups, homogeneous spaces, or more general manifolds, are described by various algebras $A$, defined over various fields…

A version of noncommutative geometry is proposed which is based on phase-space rather than position space. The momenta encode the information contained in the algebra of forms by a map which is the noncommutative extension of the duality…

We give a survey of techniques from quantum group theory which can be used to show that some quantum spaces (objects of the category dual to the category of $\mathrm{C}^*$-algebras) do not admit any quantum group structure. We also provide…

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…

Noncommutative spacetimes are widely believed to model some properties of the quantum structure of spacetime at the Planck regime. In this contribution the construction of (anti-)de Sitter noncommutative spacetimes obtained through quantum…

As is well known, an external magnetic field in configuration space coupled to a quantum dynamics induces noncommutativity in its velocity momentum space. By phase space duality, an external vector potential in the conjugate momentum sector…

Noncommutative quantum mechanics on the plane has been widely studied in the literature. Here, we consider the problem using Isham's canonical group quantization scheme for which the primary object is the symmetry group that underlies the…

We develop an approach to quantum dynamics based on quantum phase space trajectories. The latter are built from a unitary irreducible representation of the symmetry group of the respective classical phase space. We use a quantum action…

Noncommutative quantum mechanics can be considered as a first step in the construction of quantum field theory on noncommutative spaces of generic form, when the commutator between coordinates is a function of these coordinates. In this…

Non-Hermitian systems have attracted considerable interest in recent years owing to their unique topological properties that are absent in Hermitian systems. While such properties have been thoroughly characterized in free fermion models,…

Phase space is the state space of classical mechanics, and this manifold is normally endowed only with a symplectic form. The geometry of quantum mechanics is necessarily more complicated. Arguments will be given to show that augmenting the…

Building on the theory of noncommutative complex structures, the notion of a noncommutative K\"ahler structure is introduced. In the quantum homogeneous space case many of the fundamental results of classical K\"ahler geometry are shown to…