Remarks on invariant geometric calculus. Cayley-Grassmann algebras and geometric Clifford algebras

  • P. Bravi
  • A. Brini


The invariant geometric calculus was founded by the German mathematician H.G. Grassmann in 1844 (Ausdehnungslehre [15, 16]). In this treatise, he introduced the modern notion of a vector in an abstract n-dimensional space and, in general, the notion of an extensor (decomposable antisymmetric tensor). Grassmann’s plan was radically innovative; his aim was to found an intrinsic algebraic calculus for (projective, affine, euclidean) geometry, that was alternative to the cartesian idea of linking algebra and geometry by a system of coordinates. To this aim, in the Ausdehnungslehre, several algebraic operations on extensors are introduced, for example, the progressive product, the regressive product and the Ergänzung operation (in geometric language: projection, section and orthogonal duality).


Orthogonal Basis Clifford Algebra Outer Product Split Rule Hodge Operator 
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© Springer-Verlag Italia 2001

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  • P. Bravi
  • A. Brini

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