In this question I made the mistake of thinking that the alternating tensor algebra was a subalgebra of the tensor algebra. Specifically, start with $T(V)$ and consider the "antisymmetrizer" map:
$$Alt(v_1 \otimes \ldots \otimes v_n)
:=
\frac{1}{n!} \sum_{\sigma \in S_n} \text{sign}(\sigma) v_{\sigma(1)}\otimes \ldots \otimes v_{\sigma(n)}$$
The image of this map is a subspace of $T(V)$ but is not a subalgebra: for any $v_1, v_2 \in V$, we have that both are in $Alt(V)$ but $v_1 \otimes v_2$ is not.
This is easily fixed by defining a product $v_1 \hat\otimes v_2 := Alt(v_1 \otimes v_2)$. What's giving me trouble is understanding how to think of the symbol $\otimes$ in the alternating algebra, given that it's not an operation.
On one hand, we could entirely banish it, so that whenever we write an element of the algebra $Alt(T(V))$ we are only allowed to use the symbol $\hat\otimes$. On the other hand, given the definition of that symbol, it seems perfectly legitimate to say that, e.g., $v_1 \otimes v_2 – v_2 \otimes v_1$ is an element of $Alt(T(V))$. In that case $\otimes$ becomes something of an uninterpreted symbol that only has meaning in certain contexts (e.g., we cannot make sense of "$v_1 \otimes v_2$" by itself, since it's not part of the algebra).
Perhaps my question is overly philosophical, but I feel it would help if I understood how mathematicians mentally "bucket" this sort of thing.
Best Answer
I don’t think we have to make $\otimes$ a second class citizen just because it doesn’t suit the alternating tensors.
We need to keep it around so that we can unpack the definition of the wedge product sometimes.
I just think of it as an operation on the tensors under which the alternating tensors are not closed. Rather than viewing this as a cause for dispair, I think it is interesting.