Clifford Actions and the Algebra of Double Forms

Let $V$ be an oriented Euclidean vector space and let $\text{D}(V)=\mathrm{\Lambda }{V}^{\ast }\otimes \mathrm{\Lambda }{V}^{\ast }$ 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 $a\in V^{\ast }$ we introduce the inhomogeneous double form $${\chi }_a=((1\otimes a)-(a\otimes 1))e^g,$$ prove that the ${\chi }_a$ satisfy the Clifford relations for the composition product, and establish the exact regular-action formulas $$C_R(a)\omega ={\chi }_a\circ \omega ,C_L(a)\omega =-\omega \circ {\chi }_a\mathrm{.}$$ For $h\in {\text{D}}^{1,1}$ 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 ${\mathfrak{s}\mathfrak{l}}_2$ 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 $R\in {\text{D}}^{2,2}$, $${\mathrm{\Gamma }}_{1,1}(R)(\omega )=\frac{1}{4}R\mathrm{\#}\omega \mathrm{.}$$ The point of this identity is not a new definition of $R\mathrm{\#}$, 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 $\mathcal{Q}(R)=-\frac{1}{4}R\mathrm{\#}R$ satisfies $$\mathcal{Q}(R)=-{\mathrm{\Gamma }}_{1,1}(R)R,$$ and its linearization at $R$ is the operator $-2{\mathrm{\Gamma }}_{1,1}(R)$. For curvature tensors this is the sharp contribution to the Ricci-flow reaction term.