The theory of contractions of multivectors, and star duality, was reorganized in a previous article, and here we present some applications. First, we study inner and outer spaces associated to a general multivector M via the equations \(v \wedge M = 0\) and \(v \mathbin {\lrcorner }M=0\). They are then used to analyze special decompositions, factorizations and ‘carvings’ of M, to define generalized grades, and to obtain new simplicity criteria, including a reduced set of Plücker-like relations. We also discuss how contractions are related to supersymmetry, and give formulas for supercommutators of multi-fermion creation and annihilation operators.

多向量收缩理论和星对偶性在上一篇文章中进行了重新组织，在这里我们介绍一些应用。首先，我们通过方程 \（v \wedge M = 0\） 和 \（v \mathbin {\lrcorner }M=0\） 研究与一般多向量 M 相关的内部和外部空间。然后，它们用于分析 M 的特殊分解、因式分解和“雕刻”，定义广义等级，并获得新的简单性标准，包括一组简化的 Plücker 式关系。我们还讨论了收缩与超对称性的关系，并给出了多费米子产生和湮灭算子的超换向子的公式。