Introduction to Braided Geometry and $q$-Minkowski Space

Abstract

We present a systematic introduction to the geometry of linear braided spaces. These are versions of ${\mathbb{R}}^n$ in which the coordinates $x_i$ have braid-statistics described by an R-matrix. From this starting point we survey the author's braided-approach to $q$-deformation: braided differentiation, exponentials, Gaussians, integration and forms, i.e. the basic ingredients for $q$-deformed physics are covered. The braided approach includes natural $q$-Euclidean and $q$-Minkowski spaces in R-matrix form.

60 pages

References in corpus (12)

• Algebraic {$q$}-Integration and Fourier Theory on Quantum and Braided Spaces
• Quantum and Braided Linear Algebra
• q-Euclidean space and quantum group wick rotation by twisting
• Quantum Braided Groups
• *-Structures on Braided Spaces
• Braided Matrix Structure of $q$-Minkowski Space and $q$-Poincare Group
• q-Deformed Relativistic Wave Equations
• q-Epsilon tensor for quantum and braided spaces
• On the Addition of Quantum Matrices
• Lie Algebras and Braided Geometry
• Braided Supersymmetry and (CO-)HOMOLOGY
• Differential calculus on q-Minkowski space

Cited by in corpus (31)

• Quantum Groups and Noncommutative Geometry
• Field Theory on the q-deformed Fuzzy Sphere I
• Field Theory on the q-deformed Fuzzy Sphere II: Quantization
• The Small Scale Structure of Space-Time: A Bibliographical Review
• *-Structures on Braided Spaces
• Relativistic and Newtonian kappa spacetimes
• Analysis on q-deformed quantum spaces
• Deformed Minkowski spaces: clasification and properties
• Advances in Quantum and Braided Geometry
• Meaning of Noncommutative Geometry and the Planck-Scale Quantum Group
• q-Translations on quantum spaces
• Braided products for quantum spaces
• Intrinsic anyonic spin through deformed geometry
• Duality Principle and Braided Geometry
• A braided interpretation of fractional supersymmetry in higher dimensions
• Fractional spin through quatum (super)Virasoro algebras
• Some remarks on the {$q$}-Poincare algebra in R-matrix form
• Braided structure of fractional $Z_3$-supersymmetry
• Does matter differ from vacuum?
• Some Comments on Bosonisation and Biproducts
• Quantum algebras, particle phenomenology, and (quasi)supersymmetry
• An introduction to quantum groups and non-commutative differential calculus
• Fractional Spin through Quantum Affine Algebra $\hat{A}(n)$ and quantum affine superalgebra $\hat{A}(n,m)$
• On the braided Fock spaces
• Fermionic $q$-Fock Space and Braided Geometry
• Braided symmetries of $S{U}_{q,\varphi }(2)$ and Podleś Spheres
• Quantum kinematics on q-deformed quantum spaces II, Wave functions on position and momentum space
• Quantum kinematics on q-deformed quantum spaces I, Mathematical Framework
• Comments on Bosonisation and Biproducts
• Quasi-{$\ast$} Structure on {$q$}-Poincare algebras
• Non-relativistic Schroedinger theory on q-deformed quantum spaces I, Mathematical framework and equations of motion