## Journal of Generalized Lie Theory and Applications

- J. Gen. Lie Theory Appl.
- Volume 3, Number 1 (2009), Article ID S080101, 1-32.

### First-order differential calculi over multi-braided quantum groups

#### Abstract

A differential calculus of the first order over multi-braided quantum groups is developed. In analogy with the standard theory, left/right-covariant 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. These structures naturally incorporate the idea of the quantum Lie algebra associated to a given multibraded quantum group, the space of left-invariant forms corresponding to the dual of the Lie algebra itself. 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.

#### Article information

**Source**

J. Gen. Lie Theory Appl., Volume 3, Number 1 (2009), Article ID S080101, 1-32.

**Dates**

First available in Project Euclid: 19 October 2011

**Permanent link to this document**

https://projecteuclid.org/euclid.jglta/1319028472

**Digital Object Identifier**

doi:10.4303/jglta/S080101

**Mathematical Reviews number (MathSciNet)**

MR2486607

**Zentralblatt MATH identifier**

1167.58005

#### Citation

DURDEVICH, Micho. First-order differential calculi over multi-braided quantum groups. J. Gen. Lie Theory Appl. 3 (2009), no. 1, Article ID S080101, 1--32. doi:10.4303/jglta/S080101. https://projecteuclid.org/euclid.jglta/1319028472