\rho does what?

looking at the lecture notes and video, I have $\rho : V^* \otimes W^* \rightarrow (V \otimes W)^*$, and I interpret $\rho$ to take the tensor of two linear functionals and send it to one linear functional. I get confused about how to use the information that $\langle \rho (v^* \otimes w^*), v \otimes w \rangle = \langle v^*,v\rangle \langle w^*,w\rangle$.

I hoped to apply $\rho$ to $v^* \otimes w^*$ to get something “feel-good”, like $(v \otimes w)^*$, but that seems unlikely…

1. alyssa palfreyman

We don’t know explicitly what $\rho(v^*\otimes w^*)$ looks like but knowing that $\langle \rho(v^*\otimes w^*), v\otimes w\rangle = \langlev^*,v\rangle \langlew^*,w\rangle$ turns out to still be very useful. In problem 3 I was able to show that the associative and unitary diagrams commute by considering an element of $C^*$ as a linear functional from $C$ to $\mathbb{K}$ and showing how it acts on an element of $C$ (in which I applied the definition of $\rho$). Using this approach avoids the need for an explicit expression for $\rho(v^*\otimes w^*)$.

2. Brian Cruz

The cool thing is that the information we have *is* the definition of our functional! That is if

$\rho:V^{*}\otimes W^{*} \to \left(V\otimes W\right)^{*} v^{*}\otimes w^{*} \mapsto f$

where $f:V\otimes W\to\mathbb{F}$
(a functional); then for $latexc\otimes d\in V\otimes W$ we have

$f\left(c\otimes d\right)=\left[\rho\left(v^{*}\otimes w^{*}\right)\right]\left(c\otimes d\right)=\left\langle v^{*},c\right\rangle \left\langle w^{*},d\right\rangle .$

Now can we use this information to show that m
is associative? We need to show that it is for any three functionals $a^{*},b^{*},c^{*}\in C^{*}$
from the algebra (Is this what you meant Alyssa?). Remembering that our definition of m
is $\Delta^{*}\rho$
can we show that $latexm\left(a^{*},m\left(b^{*},c^{*}\right)\right)=m\left(m\left(a^{*},b^{*}\right),c^{*}\right)$
?

I used the notation

$m\left(v^{*}\otimes w^{*}\right)\left(c\right)=\left\langle m\left(v^{*}\otimes w^{*}\right),c\right\rangle$

and it really clarified it for me once I got used to it. Use the definition of m
and see if you can write your results in Sweedler notation, also using the definition of $\rho$. If you can do that then both parts of the problem become easier.

3. Brian Cruz

Sorry! Here’s what that latex really looks like that I messed up.

1st) $c\otimes d\in V\otimes W$
2nd) $m\left(a^{*},m\left(b^{*},c^{*}\right)\right)=m\left(m\left(a^{*},b^{*}\right),c^{*}\right)$