I’m utterly confused by this one:

It looks like the left-hand side lives in and the rhs lives in H. I’m assuming this: first, is a multiplication operation, and since S sends H to H, this would live in H.

I’ve tried mapping the elements, and that hasn’t helped clarify this.

What am I doing wrong here? I assume it has something to do with the relationship between and

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what I belive is,

lives in right. But it is an special element from because it lives (recall the definition of the antipode) in the image of the unit map . So the left hand really lives in so we can stablish the equality as an equivalence with respect to this congruence.

LaTeX corrected. I hate WordPress for not having a preview button…

I agree with Lisa: the formula makes no sense. The left tensorand rather clearly is considered an element living in rather than in the ground field (although it does live in the ground field as yimp34 correctly noticed), so we can’t “cancel” it.

I am not sure whether the problem was trying to say or , but I am pretty positive it is one of these.

I am totally agree with Yimp34’s response here,Lisa. From the commutative diagram,we see that can be viewed as . It is mention on Page 37 of the lecture note. Hope it helps. =)

I take back what I said above–

Well, in the last HW, in the “(Practice with Sweedler notation.)” I had a rough time trying to understand the same issue, so punctualizing the problem is simple, indeed the expression on the left lives in , and the right one is from , but we can use the product to take out the tensor expression and symplify it. Remember that Sweedler’s notation it’s only a repesentation like, fon example, even if they don’t come from the same space.

I hope this can clarify the situation a little bit.

Lisa, it’s a fair objection. Sweedler’s notation is hiding the canonical identification of with , which is itself identified with — precisely as yimp34 suggested.

In fact Sweedler’s book states this question as:

and in the next questions it just writes for . I probably should have done that too, sorry for the confusion.