The Space of Vector Fields on Q-Groups

TL;DR

This paper constructs and analyzes the space of vector fields on quantum groups, exploring their duality with 1-forms and generalizing to tensor fields within a Hopf algebra framework.

Contribution

It introduces a construction of vector fields on quantum groups, including duality with 1-forms and a generalization to tensor fields, applicable to generic Hopf algebras.

Findings

Defined vector fields as products of invariant vector fields and quantum group elements

Explored duality between vector fields and 1-forms

Proposed a Lie derivative for non-invariant vector fields

Abstract

We construct the space of vector fields on quantum groups . Its elements are products of the known left invariant vector fields with the elements of the quantum group itself. We also study the duality between vector fields and 1-forms. The construction is easily generalized to tensor fields. A Lie derivative along any (also non left invariant) vector field is proposed. These results hold for a generic Hopf algebra.