q-Epsilon tensor for quantum and braided spaces

TL;DR

This paper develops a covariant construction of the epsilon tensor, Hodge star, and differentials in braided quantum spaces like q-Euclidean and q-Minkowski, using R-matrices and quantum group symmetries.

Contribution

It introduces a general braided geometric framework for defining the epsilon tensor and related operators in quantum spaces with R-matrix structures.

Findings

Constructed the epsilon tensor for braided vector spaces.

Developed covariant Hodge star and differential operators.

Applied formalism to q-Euclidean and q-Minkowski spaces.

Abstract

The machinery of braided geometry introduced previously is used now to construct the $\epsilon$ `totally antisymmetric tensor' on a general braided vector space determined by R-matrices. This includes natural $q$-Euclidean and $q$-Minkowski spaces. The formalism is completely covariant under the corresponding quantum group such as $\widetilde{SO_q(4)}$ or $\widetilde{SO_q(1,3)}$. The Hodge $*$ operator and differentials are also constructed in this approach.