An Introduction to Noncommutative Differential Geometry and its Physical Applications: Extensions of Space-Time

https://hal.archives-ouvertes.fr/hal-01485544/document

Poisson–Lie T-duals of the bi-Yang–Baxter models

Noncommutative Spaces and Poincaré Symmetry

Classification of differential forms on κ-Minkowski space, particularly, the classification of all bicovariant differential calculi of classical dimension is presented. By imposing super-Jacobi identities we derive all possible differential algebras compatible with the κ-Minkowski algebra for time-like, space-like and light-like deformations. Embedding into the super-Heisenberg algebra is constructed using non-commutative (NC) coordinates and one-forms. Particularly, a class of differential calculi with an undeformed exterior derivative and one-forms is considered. Corresponding NC differential calculi are elaborated. Related class of new Drinfeld twists is proposed. It contains twist leading to κ-Poincare Hopf algebra for light-like deformation. Corresponding super-algebra and deformed super-Hopf algebras, as well as the symmetries of differential algebras are presented and elaborated. Using the NC differential calculus, we analyze NC field theory, modified dispersion relations, and discuss further physical applications.

/pdf/toward-the-classification-of-differential-calculi-on-k-690zsc66js.pdf

Toward the classification of differential calculi on κ-Minkowski space and related field theories

We revisit the notion of quantum Lie algebra of symmetries of a noncommutative spacetime, its elements are shown to be the generators of infinitesimal transformations and are naturally identified with physical observables. Wave equations on noncommutative spaces are derived from a quantum Hodge star operator. This general noncommutative geometry construction is then exemplified in the case of κ-Minkowski spacetime. The corresponding quantum Poincare-Weyl Lie algebra of in-finitesimal translations, rotations and dilatations is obtained. The d’Alembert wave operator coincides with the quadratic Casimir of quantum translations and it is deformed as in Deformed Special Relativity theories. Also momenta (infinitesimal quantum translations) are deformed, and correspondingly the Einstein-Planck relation and the de Broglie one. The energy-momentum relations (dispersion relations) are consequently deduced. These results complement those of the phenomenological literature on the subject.

/pdf/observables-and-dispersion-relations-in-k-minkowski-38dwkdl31z.pdf

Observables and dispersion relations in κ-Minkowski spacetime

Classification of differential forms on $\kappa$-Minkowski space, particularly, the classification of all bicovariant differential calculi of classical dimension is presented. By imposing super-Jacobi identities we derive all possible differential algebras compatible with the $\kappa$-Minkowski algebra for time-like, space-like and light-like deformations. Embedding into the super-Heisenberg algebra is constructed using non-commutative (NC) coordinates and one-forms. Particularly, a class of differential calculi with an undeformed exterior derivative and one-forms is considered. Corresponding NC differential calculi are elaborated. Related class of new Drinfeld twists is proposed. It contains twist leading to $\kappa$-Poincare Hopf algebra for light-like deformation. Corresponding super-algebra and deformed super-Hopf algebras, as well as the symmetries of differential algebras are presented and elaborated. Using the NC differential calculus, we analyze NC field theory, modified dispersion relations, and discuss further physical applications.

Toward the classification of differential calculi on $\kappa$-Minkowski space and related field theories

In this paper, we propose a simple generalization of the locally r-symmetric Jordanian twist, resulting in the one-parameter family of Jordanian twists. All the proposed twists differ by the coboundary twists and produce the same Jordanian deformation of the corresponding Lie algebra. They all provide the κ-Minkowski spacetime commutation relations. Constructions from noncommutative coordinates to the star product and coproduct, and from the star product to the coproduct and the twist are presented. The corresponding twist in the Hopf algebroid approach is given. Our results are presented symbolically by a diagram relating all of the possible constructions.

/pdf/remarks-on-simple-interpolation-between-jordanian-twists-446u6f4d7n.pdf

Remarks on simple interpolation between Jordanian twists

Realizations of $$\kappa $$
 -Minkowski space linear in momenta are studied for time-, space- and light-like deformations. We construct and classify all such linear realizations and express them in terms of the $$\mathfrak {gl}(n)$$
 generators. There are three one-parameter families of linear realizations for time-like and space-like deformations, while for light-like deformations, there are only four linear realizations. The relation between a deformed Heisenberg algebra, the star product, the coproduct of momenta, and the twist operator is presented. It is proved that for each linear realization there exists a Drinfeld twist satisfying normalization and cocycle conditions. $$\kappa $$
 -Deformed $$\mathfrak {igl}(n)$$
 -Hopf algebras are presented for all cases. The $$\kappa $$
 -Poincare–Weyl and $$\kappa $$
 -Poincare–Hopf algebras are discussed. The left–right dual $$\kappa $$
 -Minkowski algebra is constructed from the transposed twists. The corresponding realizations are nonlinear. All Drinfeld twists related to $$\kappa $$
 -Minkowski space are obtained from our construction. Finally, some physical applications are discussed.

/pdf/realizations-of-kappa-minkowski-space-drinfeld-twists-and-23gewvcb77.pdf

Danijel Pikutić

Papers

Noncommutative Spaces and Poincaré Symmetry

Toward the classification of differential calculi on κ-Minkowski space and related field theories

Toward the classification of differential calculi on $\kappa$-Minkowski space and related field theories

Remarks on simple interpolation between Jordanian twists

Realizations of $$\kappa $$-Minkowski space, Drinfeld twists, and related symmetry algebras