Noncommutative Superspaces Covariant Under $OSp_q(1|2)$ Algebra

TL;DR

This paper develops a systematic method to construct noncommutative spaces covariant under quantum groups, extending to supergroups, and introduces a family of noncommutative superspheres with potential applications in quantum geometry.

Contribution

The paper introduces a unified method for constructing covariant noncommutative spaces under quantum groups and supergroups, including a new family of noncommutative superspheres.

Findings

Unified construction of quantum plane and spheres

Extension to quantum supergroup $OSp_q(1|2)$

Introduction of covariant noncommutative superspheres

Abstract

Using the corepresentation of the quantum group $ SL_q(2)$ a general method for constructing noncommutative spaces covariant under its coaction is developed. The method allows us to treat the quantum plane and Podle\'s' quantum spheres in a unified way and to construct higher dimensional noncommutative spaces systematically. Furthermore, we extend the method to the quantum supergroup $ OSp_q(1|2).$ In particular, a one-parameter family of covariant algebras, which may be interpreted as noncommutative superspheres, is constructed.