\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…

 

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3 comments

  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)

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