Abstract

A differential calculus of the first order over multi-braided quantum groups is developed. In analogy with the standard theory, left/rightcovariant and bicovariant differential structures are introduced and investigated. Furthermore, antipodally covariant calculi are studied. The concept of the *-structure on a multi-braided quantum group is formulated, and in particular the structure of left-covariant *-covariant calculi is analyzed. A special attention is given to differential calculi covariant with respect to the action of the associated braid system. In particular it is shown that the left/right braided-covariance appears as a consequence of the left/right-covariance relative to the group action. Braided counterparts of all basic results of the standard theory are found.

Highlights

  • The basic theme of this study is the analysis of the first-order differential structures over multibraided quantum groups

  • Standard braided quantum groups are included as a special case into the theory of multi-braided quantum groups [3]

  • The difference between two types of braided quantum groups is in the behavior of the coproduct map

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Summary

Introduction

The basic theme of this study is the analysis of the first-order differential structures over multibraided quantum groups. In this context, left/right, and bi-σ-covariant differential structures are distinguished. Besides the study of properties of maps Γ and σnl , and their interrelations, we shall analyze the internal structure of left-covariant calculi. Concrete examples will be included in the part of the study, after developing a higher-order differential calculus This will include differential structures over already considered groups, as well as new examples of ‘differential’ multi-braided quantum groups coming from the developed theory. Our philosophy is that the non-trivial braidings involving the differential map should be interpreted as an extra structure given over the whole differential calculus

The concept of braided covariance
The structure of left-covariant calculi
Bicovariant calculi
Antipodally covariant calculi
A Right-covariant calculi
B Elementary properties of the adjoint action
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