Abstract
Let v be a finite dimensional vector space, dim(v) = n, and let (G(v), ∨) be its exterior algebra. We will denote by ∨ the exterior product (equivalently, the wedge or Grassmann’s progressive product) in order to stress its close analogy with its geometric lattice companion; we call this operation the join. Given two extensors (i. e. decomposable antisymmetric tensors) A and B, they represent two subspaces <A> and <B> of v), respectively.
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References
M. Barnabel, A. Brini, and G.-C. Rota, “On the exterior calculus of invariant theory.” Journal of Algebra, 1985, 96: 120–160.
J. Bokowsky and B. Sturmfels, “Polytopal and non-polytopal spheres: an algorithmic approach,” Israel J. Math., 1987, 57: 257–271.
Nicolas Bourbaki, Éléments de Mathématique, Algébre Multilinéaire ( Paris: Hermann, 1979 ).
Andrea Brini and A. G. B. Teolis, “Capelli’s method of variabili ausiliarie, superalgebras and geometric calculus,” Neil L. White (ed.), Invariant Methods in Discrete and Computational Geometry ( Dordrecht: Kluwer, 1995 ), 59–75.
Jean Dieudonné, “The tragedy of Grassmann,” Linear and Multilinear Algebra, 1979, 8: 1–14.
P. Doubilet, G.-C. Rota, and J. Stein, “On the foundations of combinatorial theory IX: Combinatorial methods in invariant theory,” Stud. Appl. Math., 1974, 53: 185–216.
Henry G. Forder, The Calculus of Extension (Cambridge: Cambridge University Press, 1940). Reprint 1960 by Chelsea Publ. Comp. NY.
F. Grosshans, G.-C. Rota, and J. Stein, “Invariant theory and superalgebras”, American Math. Society C. B. M. S. Regional Conference Series 69 (1987).
David Hestenes, Grassmann’s Vision, 1996, in this volume (1996).
David Hestenes and Renatus Ziegler, “Projective geometry with Clifford algebra,” Acta Applic. Math., 1991, 23: 25–63.
M. Marcus, Finite dimensional Multilinear Algebra, 2 vols. ( New York: Dekker, 1975 ).
Giuseppe Peano, Calcolo Geometrico secondo la Ausdehnugslehre di H. Grassmann ( Fratelli Bocca Editori: Torino, 1888 ).
Günther Pickert, Analytische Geometrie, 4. Auflage (Leipzig: Akademische Verlagsgesellschaft Geest Portig, 1961 ).
B. Sturmfels and W. Whiteley, “On the synthetic factorization of projectively invariant polynomials,” J. Symbolic Computation, 1991, 11: 439–453.
B. Segre, Lezioni sulla teoria delle forme differenziali ( Roma: Docet Editrice Universitaria, 1951 ).
M. E. Sweedler, Hopf Algebras (New York: Benjamin, 1969 ).
Hermann Weyl, The Classical Groups (Princeton: Princeton Univ. Press, 1946 ).
Neil L. White, “Multilinear Cayley factorization,” J. Symbolic Computation, 1991, 11: 421–438.
Neil L. White and W. Whiteley, “The algebraic geometry of stress in frameworks,” SIAM J. Algebraic and discrete methods, 1983, 4: 481–511.
Neil L. White and W. Whiteley, “The algebraic geometry of motions of bar-and-body frameworks,” SIAM J. Algebraic and discrete methods, 1987, 8: 1–32.
Alfred N. Whitehead, A Treatise on Universal Algebra with Applications ( Cambridge: Cambridge Univ. Press, 1898 ).
Arno Zaddach, Grassmann Algebra in der Geometrie ( Mannheim-Leipzig-Wien-Zürich: B. I. Wissenschaftsverlag, 1994 ).
Arno Zaddach, “Regressive products and Bourbaki,” in this volume (1996).
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Brini, A., Teolis, A.G.B. (1996). Grassmann Progressive and Regressive Products and CG-Algebras. In: Schubring, G. (eds) Hermann Günther Graßmann (1809–1877): Visionary Mathematician, Scientist and Neohumanist Scholar. Boston Studies in the Philosophy of Science, vol 187. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8753-2_19
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DOI: https://doi.org/10.1007/978-94-015-8753-2_19
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