Right-Covariant Differential Calculus on Hopf Superalgebra \({{\mathbb {F}}}({\mathbb {C}}_q^{2|1})\)

Abstract

We define a new \({{\mathbb {Z}}}_2\)-graded quantum (2+1)-space and show that the extended \({{\mathbb {Z}}}_2\)-graded algebra of polynomials on this \({{\mathbb {Z}}}_2\)-graded quantum space, denoted by \({\mathbb F}({{\mathbb {C}}}_q^{2\vert 1 })\), is a \({{\mathbb {Z}}}_2\)-graded Hopf algebra. We construct a right-covariant differential calculus on \({{\mathbb {F}}}({{\mathbb {C}}}_q^{2\vert 1 })\) and define a \({\mathbb Z}_2\)-graded quantum Weyl algebra and mention a few algebraic properties of this algebra. Finally, we explicitly construct the dual \({{\mathbb {Z}}}_2\)-graded Hopf algebra of \({{\mathbb {F}}}({\mathbb C}_q^{2\vert 1 })\).