Double-graded quantum superplane

Abstract

A ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded generalisation of the quantum superplane is proposed and studied. We construct a bicovariant calculus on what we shall refer to as the \emph{double-graded quantum superplane}. The commutation rules between the coordinates, their differentials and partial derivatives are explicitly given. Furthermore, we show that an extended version of the double-graded quantum superplane admits a natural Hopf ${\mathbb{Z}}_2^2$-algebra structure.

20 pages, amslatex, romp.sty, version 3: final journal version, to appear in "Reports on Mathematical Physics"