First-order Differential Calculi over Multi-braided Quantum Groups

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.