Drorbn - User contributions [en]

AKT-09/HW1

2009-10-14T05:31:37Z

Derek.goto:

{{AKT-09/Navigation}}

'''Solve the following problems''' and submit them in class by October 13, 2006:

'''Problem 1.''' If <math>f \in {\mathcal V}_n</math> and <math>g \in {\mathcal V}_m</math> then <math>f \cdot g \in {\mathcal V}_{n+m}</math> (as what one would expect by looking at degrees of polynomials) and <math>W_{f \cdot g} = m_\mathbb{Q} \circ (W_f \otimes W_g) \circ \Delta</math> where <math>(W_f \otimes W_g) \circ \Delta: {\mathcal A} \rightarrow \mathbb{Q} \otimes \mathbb{Q}</math> and <math>m_\mathbb{Q}</math> is the multiplication of rationals. (See {{AKT-09/vps|0924-2}}, minute 36:01).

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

{{Equation*|<math>P = \sum_{n=0}^\infty \frac{(-\Theta)^n}{n!}\partial_\theta^n.</math>}}

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

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

'''Idea for a good deed.''' Later than October 13, prepare a [[AKT-09/Sol1|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 [[AKT-09/Class Photo|photo page]].

[[Image:AKT-09-ClassPhoto.jpg|center|400px]]

AKT-09/Navigation

2009-10-14T05:28:37Z

Derek.goto:

{| border="1px" cellpadding="1" cellspacing="0" width="100%" style="font-size: small; align: left"
|- align=left
!#
!Week of...
!Videos, Notes, and Links
|- align=left
|align=center|1
|Sep 7
|[[AKT-09/About This Class|About This Class]] {{AKT-09/vp|0910-1}} {{AKT-09/vp|0910-2}} [[AKT-09/Tricolourability|Tricolourability]]
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|Sep 14
|{{AKT-09/vp|0915}} {{AKT-09/vp|0917-1}} {{AKT-09/vp|0917-2}}
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|Sep 21
|{{AKT-09/vp|0922}} {{AKT-09/vp|0924-1}} [[AKT-09/Class Photo|Class Photo]] {{AKT-09/vp|0924-2}}
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|align=center|4
|Sep 28
|[[AKT-09/HW1|Homework Assignment 1]] [[AKT-09/Sol1|Homework Assignment 1 Solutions]] {{AKT-09/vp|0929}} {{AKT-09/vp|1001-1}} {{AKT-09/vp|1001-2}}
|- align=left
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|Oct 5
|{{AKT-09/vp|1006}} {{AKT-09/vp|1008-1}} {{AKT-09/vp|1008-2}}
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|Oct 12
|{{AKT-09/vp|1013}} [[AKT-09/HW2|Homework Assignment 2]]
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|Oct 19
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|Oct 26
|[[AKT-09/HW3|Homework Assignment 3]]
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|Nov 2
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|Nov 9
|[[AKT-09/HW4|Homework Assignment 4]] No Thursday class.
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|Nov 16
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|Nov 23
|[[AKT-09/HW5|Homework Assignment 5]]
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|Nov 30
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|- align=left
|colspan=3 align=center|[[AKT-09/Register of Good Deeds|Register of Good Deeds]] / [[AKT-09/To Do|To Do List]]
|- align=left
|colspan=3 align=center|[[Image:AKT-09-ClassPhoto.jpg|310px]] [[AKT-09/Class Photo|Add your name / see who's in!]]
|- align=left
|colspan=3 align=center|[[Image:3x4bbs.jpg|310px]]
|}

AKT-09/Sol1

2009-10-14T05:26:25Z

Derek.goto:

As a good deed, I prepared a nice TeX writeup of the [http://katlas.math.toronto.edu/drorbn/images/1/19/Hw1_solutions.pdf first homework solutions].

If you have any comments or suggestions, please [mailto:derek.goto@gmail.com email me] (I'm taking the class remotely).

File:Hw1 solutions.pdf

2009-10-14T05:18:37Z

Derek.goto: