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We first explain the wedge product between vectors and differential forms that appears in this text. Note that we use the specific case of \(\mathbb {R}^{4,1}\) to explain these concepts, but the definitions can be easily generalized for any \(\mathbb {R}^{p,q}\) . In this text, we use the fact that the exterior algebra \(\wedge ^2 \mathbb {R}^{4,1}\) is isomorphic to the Lie algebra \(\mathfrak {o}_{4,1}\) without further comment, via \(\displaystyle \begin{aligned} \wedge^2 \mathbb{R}^{4,1} \ni A \wedge B \mapsto A \wedge B \in \mathfrak{o}_{4,1} \end{aligned}\) for some \(A,B \in \mathbb {R}^{4,1}\) , where 3.1 \(\displaystyle \begin{aligned} {} (A \wedge B)X= \langle A,X \rangle B - \langle B,X \rangle A \end{aligned} \) for any vector \(X \in \mathbb {R}^{4,1}\) . The fact that \(A \wedge B\) as in (3.1) is in the Lie algebra \(\mathfrak {o}_{4,1}\) can be directly checked:

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From Smooth to Discrete via Permutability

  • 章节

  • 发表日期：

    2025年

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  • Joseph Cho,
  • Kosuke Naokawa,
  • Yuta Ogata,
  • Mason Pember,
  • Wayne Rossman,
  • Masashi Yasumoto

摘要

We first explain the wedge product between vectors and differential forms that appears in this text. Note that we use the specific case of \(\mathbb {R}^{4,1}\) to explain these concepts, but the definitions can be easily generalized for any \(\mathbb {R}^{p,q}\) . In this text, we use the fact that the exterior algebra \(\wedge ^2 \mathbb {R}^{4,1}\) is isomorphic to the Lie algebra \(\mathfrak {o}_{4,1}\) without further comment, via \(\displaystyle \begin{aligned} \wedge^2 \mathbb{R}^{4,1} \ni A \wedge B \mapsto A \wedge B \in \mathfrak{o}_{4,1} \end{aligned}\) for some \(A,B \in \mathbb {R}^{4,1}\) , where 3.1 \(\displaystyle \begin{aligned} {} (A \wedge B)X= \langle A,X \rangle B - \langle B,X \rangle A \end{aligned} \) for any vector \(X \in \mathbb {R}^{4,1}\) . The fact that \(A \wedge B\) as in (3.1) is in the Lie algebra \(\mathfrak {o}_{4,1}\) can be directly checked: