# AKT-09/HW1

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Solve the following problems and submit them in class by October 13, 2009:

Problem 1. If ${\displaystyle f\in {\mathcal {V}}_{n}}$ and ${\displaystyle g\in {\mathcal {V}}_{m}}$ then ${\displaystyle f\cdot g\in {\mathcal {V}}_{n+m}}$ (as what one would expect by looking at degrees of polynomials) and ${\displaystyle W_{f\cdot g}=m_{\mathbb {Q} }\circ (W_{f}\otimes W_{g})\circ \Box }$ where ${\displaystyle (W_{f}\otimes W_{g})\circ \Box :{\mathcal {A}}\rightarrow \mathbb {Q} \otimes \mathbb {Q} }$ and ${\displaystyle m_{\mathbb {Q} }}$ is the multiplication of rationals. (See 090924-2, minute 36:01).

Problem 2. Let ${\displaystyle \Theta :{\mathcal {A}}\to {\mathcal {A}}}$ be the multiplication operator by the 1-chord diagram ${\displaystyle \theta }$, and let ${\displaystyle \partial _{\theta }={\frac {d}{d\theta }}}$ be the adjoint of multiplication by ${\displaystyle W_{\theta }}$ on ${\displaystyle {\mathcal {A}}^{\star }}$, where ${\displaystyle W_{\theta }}$ is the obvious dual of ${\displaystyle \theta }$ in ${\displaystyle {\mathcal {A}}^{\star }}$. Let ${\displaystyle P:{\mathcal {A}}\to {\mathcal {A}}}$ be defined by

${\displaystyle P=\sum _{n=0}^{\infty }{\frac {(-\Theta )^{n}}{n!}}\partial _{\theta }^{n}.}$

Verify the following assertions, but submit only your work on assertions 4,5,7,11:

1. ${\displaystyle \left[\partial _{\theta },\Theta \right]=1}$, where ${\displaystyle 1:{\mathcal {A}}\to {\mathcal {A}}}$ is the identity map and where ${\displaystyle [A,B]:=AB-BA}$ for any two operators.
2. ${\displaystyle P}$ is a degree ${\displaystyle 0}$ operator; that is, ${\displaystyle \deg Pa=\deg a}$ for all ${\displaystyle a\in {\mathcal {A}}}$.
3. ${\displaystyle \partial _{\theta }}$ satisfies Leibnitz' law: ${\displaystyle \partial _{\theta }(ab)=(\partial _{\theta }a)b+a(\partial _{\theta }b)}$ for any ${\displaystyle a,b\in {\mathcal {A}}}$.
4. ${\displaystyle P}$ is an algebra morphism: ${\displaystyle P1=1}$ and ${\displaystyle P(ab)=(Pa)(Pb)}$.
5. ${\displaystyle \Theta }$ satisfies the co-Leibnitz law: ${\displaystyle \Box \circ \Theta =(\Theta \otimes 1+1\otimes \Theta )\circ \Box }$ (why does this deserve the name "the co-Leibnitz law"?).
6. ${\displaystyle P}$ is a co-algebra morphism: ${\displaystyle \eta \circ P=\eta }$ (where ${\displaystyle \eta }$ is the co-unit of ${\displaystyle {\mathcal {A}}}$) and ${\displaystyle \Box \circ P=(P\otimes P)\circ \Box }$.
7. ${\displaystyle P\theta =0}$ and hence ${\displaystyle P\langle \theta \rangle =0}$, where ${\displaystyle \langle \theta \rangle }$ is the ideal generated by ${\displaystyle \theta }$ in the algebra ${\displaystyle {\mathcal {A}}}$.
8. If ${\displaystyle Q:{\mathcal {A}}\to {\mathcal {A}}}$ is defined by
${\displaystyle Q=\sum _{n=0}^{\infty }{\frac {(-\Theta )^{n}}{(n+1)!}}\partial _{\theta }^{(n+1)}}$
then ${\displaystyle a=\theta Qa+Pa}$ for all ${\displaystyle a\in {\mathcal {A}}}$.
9. ${\displaystyle \ker P=\langle \theta \rangle }$.
10. ${\displaystyle P}$ descends to a Hopf algebra morphism ${\displaystyle {\mathcal {A}}^{r}\to {\mathcal {A}}}$, and if ${\displaystyle \pi :{\mathcal {A}}\to {\mathcal {A}}^{r}}$ is the obvious projection, then ${\displaystyle \pi \circ P}$ is the identity of ${\displaystyle {\mathcal {A}}^{r}}$. (Recall that ${\displaystyle {\mathcal {A}}^{r}={\mathcal {A}}/\langle \theta \rangle }$).
11. ${\displaystyle P^{2}=P}$.

Idea for a good deed. Later than October 13, prepare a beautiful TeX writeup (including the motivation and all the details) of the solution of this assignment for publication on the web. For all I know this information in this form is not available elsewhere.

Mandatory but unenforced. Find yourself in the class photo and identify yourself as explained in the photo page.