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

TL;DR

This paper classifies all bicovariant differential calculi on $kappa$-Minkowski space, explores their algebraic structures, and discusses implications for non-commutative field theories and symmetries.

Contribution

It provides a comprehensive classification of differential calculi compatible with $kappa$-Minkowski space, including new Drinfeld twists and their physical applications.

Findings

Classified all bicovariant differential calculi on $kappa$-Minkowski space.

Constructed embeddings into super-Heisenberg algebra using NC coordinates.

Proposed new Drinfeld twists leading to $kappa$-Poincaré Hopf algebra.

Abstract

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 $\kappa$-Poincar\'e 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.