Names for the vector spaces $T(V)$ and $S (V)$

Question

Are there any names for the vector spaces $T(V) = \bigoplus_{n\geq 0} V^{\otimes n}$ and $S(V)= \bigoplus_{n\geq 0} V^{\otimes n}/\Sigma_n$?

The best thing I could come up with is "the underlying vector space of the tensor/symmetric algebra on $V$"

Answer 1

I've yet to see a specific name given to $\operatorname{T}(V)$ or $\operatorname{S}(V)$ regarded just as vector spaces. So yeah, referring to them as "the underlying vector space of the tensor/symmetric algebra on $V$" is just fine. That phrase makes it clear what you are talking about, and that's really all that matters.

Or maybe here's an alternative idea. Looking just at $\operatorname{T}(V)$, when you construct the object $\operatorname{T}(V) = \oplus_{n \geq 0} V^{\otimes n}$, you're really only building a set. This set only ends up being called the tensor algebra because you're imposing an algebra structure on it. You could also impose a coalgebra structure on $\operatorname{T}(V)$, and then you would call it the tensor coalgebra. So with this reasoning, if you only want to consider the vector space structure on $\operatorname{T}(V)$, you should call is the tensor vector space of $V$. But that would probably be confusing to readers, since $V$ is a vector space too. And then all of this similarly applies to the symmetric algebra/coalgebra/vector space.

Another alternative idea would be to take this route that is notationally silly but technically correct: Say you're working over a field $k$. Define the forgetful functor $F$ from the category $\mathsf{Ass}_k$ of associative algebras over $k$ to the category $\mathsf{Vect}_k$ of vector spaces over $k$. Then the underlying vector space of the tensor algebra can be written as $F\operatorname{T}(V)$.