Realization of Vector fields for Quantum Groups as Pseudodifferential Operators on Quantum Spaces

TL;DR

This paper constructs vector fields for quantum groups as pseudodifferential operators on quantum spaces, revealing their algebraic properties and characteristic identities, with detailed analysis of real forms.

Contribution

It provides explicit, simple expressions for quantum group vector fields as pseudodifferential operators, advancing understanding of their algebraic structure.

Findings

Vector fields expressed as pseudodifferential operators

Satisfaction of characteristic polynomial identities

Analysis of real forms $SU_q(N)$ and $SO_q(N,R)$

Abstract

The vector fields of the quantum Lie algebra are described for the quantum groups $GL_q(N), SL_q(N)$ and $SO_q(N)$ as pseudodifferential operators on the linear quantum spaces covariant under the corresponding quantum group. Their expressions are simple and compact. It is pointed out that these vector fields satisfy certain characteristic polynomial identities. The real forms $SU_q(N)$ and $SO_q(N,R)$ are discussed in detail.