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 \({{\mathbb {F}}}({\mathbb C}_q^{2\vert 1 })\).

我们定义一个新的\({{\mathbb {Z}}}_2\)分级量子 (2+1) 空间并证明扩展的\({{\mathbb {Z}}}_2\)分级代数此\({{\mathbb {Z}}}_2\)分级量子空间上的多项式，表示为\({\mathbb F}({{\mathbb {C}}}_q^{2\vert 1 } )\)是一个\({{\mathbb {Z}}}_2\)分级的 Hopf 代数。我们在\({{\mathbb {F}}}({{\mathbb {C}}}_q^{2\vert 1 })\)上构造右协变微积分，并定义一个\({\mathbb Z }_2\)分级量子韦尔代数并提及该代数的一些代数性质。最后，我们明确构造\({{\mathbb {F}}}({\mathbb C}_q^{2\vert 1) 的对偶 \({{\mathbb {Z }}}_2\)分级 Hopf 代数})\)。