looking at the lecture notes and video, I have , and I interpret 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 .

I hoped to apply to to get something “feel-good”, like , but that seems unlikely…

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We don’t know explicitly what looks like but knowing that 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 as a linear functional from to and showing how it acts on an element of (in which I applied the definition of ). Using this approach avoids the need for an explicit expression for .

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

where

(a functional); then for $latexc\otimes d\in V\otimes W $ we have

Now can we use this information to show that m

is associative? We need to show that it is for any three functionals

from the algebra (Is this what you meant Alyssa?). Remembering that our definition of m

is

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

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 . If you can do that then both parts of the problem become easier.

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

1st)

2nd)