Whitney algebras and Grassmann's regressive products
type: Item
instance of: scholarly article
title: Whitney algebras and Grassmann's regressive products
zbMATH Open document ID: 1251.15029
author: Francesco Regonati
author: Andrea Brini
published in: Journal of Geometry
publication date: 2012-04-12T00:00:00Z
full work available at URL: https://arxiv.org/abs/1010.2964
review text: 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.
Mathematics Subject Classification ID: 15A75
Mathematics Subject Classification ID: 05E10
Mathematics Subject Classification ID: 15A72
Mathematics Subject Classification ID: 05B35
zbMATH DE Number: 6023277
zbMATH Keywords: exterior algebra
zbMATH Keywords: Cayley-Grassmann algebra
zbMATH Keywords: Hopf algebra
zbMATH Keywords: Whitney algebra
zbMATH Keywords: polarization operator
zbMATH Keywords: straightening law
zbMATH Keywords: letterplace algebra
zbMATH Keywords: modular lattice
zbMATH Keywords: Peano space
zbMATH Keywords: alternating multilinear form
zbMATH Keywords: bracket
zbMATH Keywords: matroid
zbMATH Keywords: Hodge star operator
MaRDI profile type: Publication
OpenAlex ID: W2004005882
arXiv ID: 1010.2964
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DOI: 10.1007/S00022-011-0081-3
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description: scientific article; zbMATH DE number 6023277