This application can be generalized to polynomials in n variables, using the Exterior algebra of an n-dimensional vector space.
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In particular, the exterior derivative gives the exterior algebra of differential forms on a manifold the structure of a differential algebra.
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The infinitesimal differentials of single variable calculus become differential forms in multivariate calculus, and their manipulation is done with exterior algebra.
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Examples of this are the exterior algebra, the symmetric algebra, Clifford algebras, the Weyl algebra and universal enveloping algebras.
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Stated like this, the dichotomy makes no reference to the metric " ? ", only to exterior algebra.
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An exterior algebra is an example of a graded-commutative ring that is not commutative in the non-graded sense.
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Quantum group ) of the exterior algebra, in the same way that the Weyl algebra is a quantization of the symmetric algebra.
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So we can apply the fact of exterior algebra that there is a unique linear form L on \ Lambda ^ 2 V that
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With an "'antipode "'defined on homogeneous elements by, the exterior algebra is furthermore a Hopf algebra.
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:: That exterior algebra thing is old, when I was very new to physics, as I've said before.
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