AKT-09/HW1 - Revision history

Drorbn at 17:00, 19 October 2009

2009-10-19T17:00:55Z

← Older revision		Revision as of 13:00, 19 October 2009
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	'''Solve the following problems''' and submit them in class by October 13, 2009:		'''Solve the following problems''' and submit them in class by October 13, 2009:

	'''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 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 \Box</math> where <math>(W_f \otimes W_g) \circ \Box: {\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		'''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

Drorbn at 16:40, 19 October 2009

2009-10-19T16:40:55Z

← Older revision		Revision as of 12:40, 19 October 2009
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	{{AKT-09/Navigation}}		{{AKT-09/Navigation}}

	'''Solve the following problems''' and submit them in class by October 13, ~~2006~~:		'''Solve the following problems''' and submit them in class by October 13, 2009:

	'''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 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).

Derek.goto at 05:31, 14 October 2009

2009-10-14T05:31:37Z

← Older revision		Revision as of 01:31, 14 October 2009
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	# <math>P^2=P</math>.		# <math>P^2=P</math>.

	'''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.		'''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]].		'''Mandatory but unenforced.''' Find yourself in the class photo and identify yourself as explained in the [[AKT-09/Class Photo\|photo page]].

Drorbn at 22:34, 28 September 2009

2009-09-28T22:34:39Z

← Older revision		Revision as of 18:34, 28 September 2009
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	# <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</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>.		# <math>P^2=P</math>.

			'''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 [[AKT-09/Class Photo\|photo page]].		'''Mandatory but unenforced.''' Find yourself in the class photo and identify yourself as explained in the [[AKT-09/Class Photo\|photo page]].

Drorbn at 22:14, 28 September 2009

2009-09-28T22:14:19Z

← Older revision		Revision as of 18:14, 28 September 2009
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	{{AKT-09/Navigation}}		{{AKT-09/Navigation}}
	{{In Preparation}}

	'''Solve the following problems''' and submit them in class by October 13, 2006:		'''Solve the following problems''' and submit them in class by October 13, 2006:
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	{{Equation*\|<math>P = \sum_{n=0}^\infty \frac{(-\Theta)^n}{n!}\partial_\theta^n.</math>}}		{{Equation*\|<math>P = \sum_{n=0}^\infty \frac{(-\Theta)^n}{n!}\partial_\theta^n.</math>}}

	~~The~~ following assertions ~~can~~ be ~~verified~~:		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>\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.
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	# <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>\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>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>\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</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>.		# <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>.		# 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>\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</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>.		# <math>P^2=P</math>.

Drorbn at 22:00, 28 September 2009

2009-09-28T22:00:02Z

New page

{{AKT-09/Navigation}}
{{In Preparation}}

'''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>}}

The following assertions can be verified:

# <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>.

'''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]]