Whitney代数与Grassmann回归积
Whitney algebras and Grassmann's regressive products
摘要：
几何产品的张量幂$Λ(V)^{\otimes m}$的外部代数和惠特尼代数\cite{crasch}提供了严格的版本格拉斯曼的{ \it回归产品}的1844年\cite{gra1}。我们研究了几何积及其与外代数上其他经典算子的关系，如Hodge $\ast-$算子和Cayley-Grassmann代数\cite{BBR, Stew}中的{ \it join}和{ \it meet}积.我们建立了张量幂$Λ(V)^{\otimes m}$和Whitney代数$W^m(M)$的字母代数的编码和它们的几何积的极化算子的除幂的编码。我们使用这些编码提供简单的证明的Crapo和施密特交换关系的惠特尼代数和两个典型的身份类别的Cayley-Grassmann代数。
英文摘要：
Geometric products on tensor powers $Λ(V)^{\otimes m}$ of an exterior algebra and on Whitney algebras \cite{crasch} provide a rigorous version of Grassmann's {\it regressive products} of 1844 \cite{gra1}. We study geometric products and their relations with other classical operators on exterior algebras, such as the Hodge $\ast-$operators and the {\it join} and {\it meet} products in Cayley-Grassmann algebras \cite{BBR, Stew}. We establish encodings of tensor powers $Λ(V)^{\otimes m}$ and of Whitney algebras $W^m(M)$ in terms of letterplace algebras and of their geometric products in terms of divided powers of polarization operators. We use these encodings to provide simple proofs of the Crapo and Schmitt exchange relations in Whitney algebras and of two typical classes of identities in Cayley-Grassmann algebras.
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