Domain of the wedge product in Little Spivak

Question

Hello! in Little Spivak, p. 79, we find this:

We will therefore define a new product, the wedge product $\omega\wedge\eta\in > \Lambda^{k+\ell}(V)$ by $$ > \omega\wedge\eta = > \frac{(k+\ell)!}{k!\ell!}\mathrm{Alt}(\omega\otimes\eta). > $$

It is ambiguous whether Spivak means for this to be defined for all $\omega\in\mathcal{J}^k(V),\eta\in\mathcal{J}^{\ell}(V)$ or only for $\omega\in\Lambda^k(V),\eta\in\Lambda^{\ell}(V)$. ($\mathcal{J}^n(V)$ is the set of $n$-tensors over vector space $V$, $\Lambda^n(V)$ is the set of alternating $n$-tensors over $V$.) Certainly the definition would make sense either way.

Which one is it?

In short: Is the wedge product applied to general tensors, or only to tensors which are already antisymmetric?

Answer 1

Without having the book handy, I can't say for sure what Spivak means by this. As KConrad points out in his comment, this is usually applied to tensors that are already alternating, and it turns the vector space $$ \Lambda(V) := \bigoplus_{k=0}^\infty \Lambda^k(V) $$ into an associative algebra, called the exterior algebra of $V$.

I would argue that this is sort of an unnatural way to view the exterior algebra. See this question, where this viewpoint is discussed. The gist of it is the following: the exterior algebra is most naturally viewed as the quotient algebra $T(V)/\langle x \otimes y + y \otimes x \mid x,y \in V\rangle$, where $T(V)$ is the tensor algebra of $V$. The description above of $\Lambda(V)$ as the direct sum of the antisymmetric tensors means that we are embedding the exterior algebra as a subspace of the tensor algebra $T(V)$ rather than a quotient. It is not a sub-algebra, because the concatenation $\omega \otimes \eta$ of two antisymmetric tensors is not antisymmetric. So the definition you/Spivak give of the wedge product is the way to express the natural multiplication of the exterior algebra in terms of operations in the tensor algebra. Applying $\mathrm{Alt}$ makes it antisymmetric again, and the factorials are a fudge factor required to make it associative.