Quantum Groups and Noncommutative Geometry

Abstract

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 powerful enough to make sense in the quantum domain. Just as the last century saw the birth of classical geometry, so the present century sees at its end the birth of this quantum or noncommutative geometry, both as an elegant mathematical reality and in the form of the first theoretical predictions for Planck-scale physics via ongoing astronomical measurements. Noncommutativity of spacetime, in particular, amounts to a postulated new force or physical effect called cogravity.

arXiv comment72 pages, many figures; intended for wider theoretical physics community (special millenium volume of JMP)

References in corpus (22)

• Hopf Algebras, Renormalization and Noncommutative Geometry
• Bicrossproduct structure of $\kappa$-Poincare group and non-commutative geometry
• Hopf algebras, cyclic cohomology and the transverse index theory
• Dual Non-Abelian Duality and the Drinfeld Double
• Quantum group gauge theory on quantum spaces
• Braided Momentum Structure of the q-Poincare Group
• Coalgebra Gauge Theory
• Quantum deformation of quantum gravity
• Algebraic {$q$}-Integration and Fourier Theory on Quantum and Braided Spaces
• Twisting of quantum differentials and the Planck scale Hopf algebra
• Free Braided Differential Calculus, Braided Binomial Theorem and the Braided Exponential Map
• q-Euclidean space and quantum group wick rotation by twisting
• Projective module description of the q-monopole
• Quantum and Braided Lie Algebras
• Introduction to Braided Geometry and $q$-Minkowski Space
• *-Structures on Braided Spaces
• Braided Matrix Structure of $q$-Minkowski Space and $q$-Poincare Group
• q-Epsilon tensor for quantum and braided spaces
• Double-Bosonisation and the Construction of {$U_q(g)$}
• Braided Geometry of the Conformal Algebra
• Duality Principle and Braided Geometry
• Quantum and braided diffeomorphism groups

Cited by in corpus (37)

• Spacetime geometry from algebra: spin foam models for non-perturbative quantum gravity
• Noncommutative geometry of angular momentum space U(su(2))
• Towards classical geometrodynamics from Group Field Theory hydrodynamics
• Spacetime Curvature in terms of Scalar Field Propagators
• kappa-Minkowski representations on Hilbert spaces
• The Momentum Spaces of $\kappa$-Minkowski noncommutative spacetime
• Gauge Theories on Deformed Spaces
• Magnetic operations: a little fuzzy physics?
• Quantum group covariant (anti)symmetrizers, epsilon-tensors, vielbein, Hodge map and Laplacian
• Emergence of Geometric phase shift in Planar Non-commutative Quantum Mechanics
• $\kappa$-Poincaré-comodules, Braided Tensor Products and Noncommutative Quantum Field Theory
• Fusing the coordinates of quantum superspace
• Dynamical noncommutativity
• The Podles spheres converge to the sphere
• Universality of Tsallis Non-Extensive Statistics and Fractal Dynamics for Complex Systems
• Noncommutative Unification of General Relativity and Quantum Mechanics. A Finite Model
• Effect of dynamical noncommutativity on the limiting mass of white dwarfs
• Coherent States for 3d Deformed Special Relativity: semi-classical points in a quantum flat spacetime
• Extending general covariance: Moyal-type noncommutative manifolds
• The noncommutative geometry of the discrete Heisenberg group
• Emergent entropy of exotic oscillators and squeezing in three-wave mixing process
• $\kappa$-Deformation and Spectral Triples
• Covariant Differential Calculi on $S{P}_q^{1|2}$
• Noncommutative Algebra and Representation Theory: Symmetry, Structure & Invariants
• Multiparticle states in braided lightlike $\kappa$-Minkowski noncommutative QFT
• Understanding the Multi-Scale and Multi-fractal Dynamics of Space Plasmas Through Tsallis Non-Extensive Statistical Theory
• Covariant formulation of Noether's Theorem for translations on kappa-Minkowski spacetime
• Fredholm modules over certain group C*-algebras
• Tsallis Non-Extensive Statistics. Theory and Applications
• Band topology of pseudo-Hermitian phases through tensor Berry connections and quantum metric
• Cartan-Weyl Basis for Yangian Double $DY(s{l}_3)$
• Quantum Groups and Polymer Quantum Mechanics
• Quantum-sl(2) action on a divided-power quantum plane at even roots of unity
• Infinities as a measure of our ignorance
• Operadic deformations as a tool for cogravity
• K-homology of certain group C*-algebras
• Isometric Spectral Subtriples