There are a few graded algebras we can construct with our symmetric and antisymmetric tensors, and at least one of them will be useful. Remember that we also have symmetric and alternating multilinear functionals in play, so the same constructions will give rise to even more algebras.
First and easiest we have the tensor algebra on . This just takes all the tensor powers of and direct sums them up
This gives us a big vector space — an infinite-dimensional one, in fact — but it’s not an algebra until we define a bilinear multiplication. For this one, we’ll just define the multiplication by the tensor product itself. That is, if and are two tensors, their product will be , which is by definition bilinear. This algebra has an obvious grading by the number of tensorands.
This is exactly the free algebra on a vector space, and it’s just like we built the free ring on an abelian group. If we perform the construction on the dual space we get an algebra of functions. If has dimension , then this is isomorphic to the algebra of noncommutative polynomials in variables.
Next we consider the symmetric algebra on , which consists of the direct sum of all the spaces of symmetric tensors
with a grading again given by the number of tensorands.
Now, despite the fact that each is a subspace of the tensor space , this is not a subalgebra of . This is because the tensor product of two symmetric tensors may well not be symmetric itself. Instead, we will take the tensor product of and , and then symmetrize it, to give . This will be bilinear, and it will work with our choice of grading, but will it be associative?
If we have three symmetric tensors , , and , then we could multiply them by or by . To get the first of these, we tensor and , symmetrize the result, then tensor with and symmetrize that. But since symmetrizing consists of adding up a number of shuffled versions of this tensor, we could tensor with first and then symmetrize only the first tensorands, before finally tensoring the entire thing. I assert that these two symmetrizations — the first one on only part of the whole term — are equivalent to simply symmetrizing the whole thing. Similarly, symmetrizing the last tensorands followed by symmetrizing the whole thing is equivalent to just symmetrizing the whole thing. And so both orders of multiplication are the same, and the operation indeed defines an associative multiplication.
To see this, remember that symmetrizing the whole term involves a sum over the symmetric group , while symmetrizing over the beginning involves a sum over the subgroup consisting of those permutations acting on only the first places. This will be key to our proof. We consider the collection of left cosets of within . For each one, we can pick a representative element (this is no trouble since there are only a finite number of cosets with a finite number of elements each) and collect these representatives into a set . Then the whole group is the disjoint union
This will let us rewrite the symmetrizer in such a way as to make our point. So let’s write down the product of the two group algebra elements we’re interested in
Essentially, because the symmetrization of the whole term subsumes symmetrization of the first tensorands, the smaller symmetrization can be folded in, and the resulting sum counts the whole sum exactly times, which cancels out the normalization factor. And this proves that the multiplication is, indeed, associative.
This multiplication is also commutative. Indeed, given and , we can let be the permutation which moves the last slots to the beginning of the term and the first slots to the end. Then we write
because right-multiplication by just shuffles around the order of the sum.
The symmetric algebra is the free commutative algebra on the vector space . And so it should be no surprise that the symmetric algebra on the dual space is isomorphic to the algebra of polynomial functions on , where the grading is the total degree of a monomial. If has finite dimension , we have .