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Grassmann algebra

A Grassmann algebra (also known as an exterior algebra) is a unital associative algebra K generated by a set, S subject to the relation χξ+ξχ=0 for any χ,ξ in S. This definition amounts to saying that the generators are anti-commuting quantities (and otherwise 'as general as possible); it should be modified in case K has characteristic 2.

The construction of such an algebra comes from the wedge product: take the vector space V that has S as basis, and the direct sum of all the exterior powers of V, using wedge product in each graded piece. If S is finite of cardinality n, the Grassmann algebra has as basis one wedge product for each subset of S, and each product made up by wedging elements of S with repeats is equal to 0.

See Fermion, Supersymmetry, Superspace, Superalgebra, Supergroup, Hermann Grassmann, p-form, Berezin integral

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