# AKT-09/HW1

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

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

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

$P = \sum_{n=0}^\infty \frac{(-\Theta)^n}{n!}\partial_\theta^n.$

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

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

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.