Whitney algebras and Grassmann's regressive products
This paper presents a discussion of three algebras that stem from the ideas of Grassmann and Peano, and developed by Rota and his school -- see, for example, \textit{M. Barnabei, A. Brini} and \textit{G.-C. Rota}, [``On the exterior calculus of invariant theory'', J. Algebra 96, 120--160 (1985; Zbl 0585.15005)]. They are: (i) the Cayley-Grassmann (CG) algebra of a Peano space -- that is, a vector space endowed with a nondegenerate alternating multilinear form (the ``bracket'') -- where the join and meet products are defined; (ii) the tensor powers of the exterior algebra of a vector space endowed with geometric products; and (iii) the Whitney algebras of a matroid, endowed with geometric products. Particular attention is given to the discussion of the geometric products in the tensor powers of the exterior algebra and their relations with other classical operators such as the Hodge star operator and the join and meet products in CG algebras. The paper also establishes encodings of the tensor powers and of Whitney algebras in terms of letterplace algebras and of the their geometric products in terms of divided powers of polarization operators. These encodings are used to prove some results in Whitney and CG algebras.
scientific article; zbMATH DE number 6023277
scientific article