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 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.

中文翻译：

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双梯度量子超平面

提出并研究了量子超平面的 $\mathbb{Z}_2 \times \mathbb{Z}_2$ 分级泛化。我们在所谓的双梯度量子超平面上构建了一个双协变微积分。明确给出了坐标、它们的微分和偏导数之间的交换规则。此外，我们证明了双梯度量子超平面的扩展版本允许自然的 Hopf $\mathbb{Z}_2^2$-代数结构。