History of Tensor/Wedge

	Revision 2 . . October 2, 2001 11:52 pm by AxelBoldt [Moved to Tensor product, which needs to be cleaned up.]
	Revision 1 . . October 2, 2001 5:31 pm by (logged).115.22.xxx [first edit]

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Changed: 1,38c1

The Tensor space is useful when wishing to deal with bilinear operators as if they were linear operators. However, linear subspaces of bilinear operators (or in general, multilinear operators) determine natural quotient spaces of the Tensor space, which are frequently useful. The definition of an anti-symmetric multilinear operator is an operator m: Vn -> X such that if there is a linear dependance between its arguments, the result is 0. The most famous example of an anti-symmetric operator is the determinant. The nth wedge space W, for a module V over a commutative ring R, together with the anti-symmetric linear wedge operator w: Vn -> W is such that for every n-linear anti-symmetric operator m: Vn -> X there exists a unique linear operator l: W -> X such that m = l o w. The wedge is unique up to a unique isomorphism. One way of defining the wedge space constructively is by dividing the Tensor space by the subspace generated by all the tensors of n-tuples which are linearily dependant. The dimension of the kth wedge space for a free module of dimension n is n!/k!(n-k)!. In particular, that means that up to a constant, there is a single anti-symmetric functional with the arity of the dimension of the space. Also note that every linear functional is anti-symmetric. Note that the wedge operator commutes with the * operator. In other words, we can define a wedge on functionals such that the result is an anti-symmetric multilinear functional. In general, we can define the wedge of an n-linear anti-symmetric functional and an m-linear anti-symmetric functional to be an (n+m)-linear anti-symmetric functional. Since it turns out that this operation is associative, we can also define the power of an anti-symmetric linear functional. When dealing with differentiable manifolds, we define a "n-form" to be a function from the manifold to the n-th wedge of the cotangent bundle. Such a form will be said to be differentiable if, when applied to n differentiable vector fields, the result is a differentiable function.