Clifford 作用与双形式代数
Clifford Actions and the Algebra of Double Forms
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中文总结 AI 辅助
本文在 Greub 复合代数中实现了双形式上的两个交换 Clifford 作用，通过构造满足 Clifford 关系的非齐次双形式，得到左-右作用的正则公式及四分量分解，并应用于曲率张量的 Ricci 流反应项。
中文摘要 AI 辅助
设 $V$ 为定向欧氏向量空间，$\D(V)=\Lambda V^*\otimes\Lambda V^*$ 为双形式代数。两个外因子承载自然的逐因子外乘与内乘算子，因此存在两个交换的 Clifford 作用。本文的核心结果是在 Greub 复合代数内部给出这些 Clifford 作用的内在实现。对 $a\in V^*$，我们引入非齐次双形式 \\[ \chi_a=((1\otimes a)-(a\otimes1))e^g, \\] 证明 $\chi_a$ 对复合乘积满足 Clifford 关系，并建立精确的正则作用公式 \\[ C_R(a)\omega=\chi_a\circ\omega, \qquad C_L(a)\omega=-\omega\circ\chi_a. \\] 对 $h\in\D^{1,1}$，这给出耦合的左-右 Clifford 作用的一个典型四分量分解，分解为外乘、双收缩以及两个广义 Bianchi 算子。对于度量双形式 $g$，这四个算子生成两个交换的 $\mathfrak{sl}_2$ 作用。该构造典范地推广到任意双次数。在双次数 $(2,2)$ 中，其保次数分量可与 sharp 积的双形式推广直接比较：对每个 $R\in\D^{2,2}$，\\[ \Gamma_{1,1}(R)(\omega)=\frac14R\\#\omega. \\] 此恒等式的要点不在于对 $R\\#$ 的新定义（它已由 Clifford 换位子构造），而在于它典范地作为高次左-右 Clifford 变换的中心齐次分量出现。作为应用，代数二次映射 $\mathcal Q(R)=-\frac14R\\#R$ 满足 \\[ \mathcal Q(R)=-\Gamma_{1,1}(R)R, \\] 且其在 $R$ 处的线性化为算子 $-2\Gamma_{1,1}(R)$。对于曲率张量，这正是 Ricci 流反应项中的 sharp 贡献。
英文摘要
Let $V$ be an oriented Euclidean vector space and let $\D(V)=ΛV^*\otimesΛV^*$ be the algebra of double forms. The two exterior factors carry natural factorwise exterior and interior multiplication operators and therefore two commuting Clifford actions. The central result of the paper is an intrinsic realization of these Clifford actions inside Greub's composition algebra. For $a\in V^*$ we introduce the inhomogeneous double form \[ χ_a=((1\otimes a)-(a\otimes1))e^g, \] prove that the $χ_a$ satisfy the Clifford relations for the composition product, and establish the exact regular-action formulas \[ C_R(a)ω=χ_a\circω, \qquad C_L(a)ω=-ω\circχ_a. \] For $h\in\D^{1,1}$ this yields a canonical four-component decomposition of the coupled left--right Clifford action into exterior multiplication, double contraction and two generalized Bianchi operators. For the metric double form $g$ these four operators generate two commuting $\mathfrak{sl}_2$ actions. The construction extends canonically to arbitrary bidegree. In bidegree $(2,2)$ its degree-preserving component admits a direct comparison with the double-form extension of the sharp product: for every $R\in\D^{2,2}$, \[ Γ_{1,1}(R)(ω)=\frac14R\#ω. \] The point of this identity is not a new definition of $R\#$, which is already constructed from Clifford commutators, but the fact that it appears canonically as the central homogeneous component of the higher left--right Clifford transform. As an application, the algebraic quadratic map $\mathcal Q(R)=-\frac14R\#R$ satisfies \[ \mathcal Q(R)=-Γ_{1,1}(R)R, \] and its linearization at $R$ is the operator $-2Γ_{1,1}(R)$. For curvature tensors this is the sharp contribution to the Ricci-flow reaction term.
发表机构
- American International University (AIU)（美国国际大学）
机构由 AI 辅助整理，请以论文原文为准。
