Difference between revisions of "AKT-09/HW1"
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− | '''Solve the following problems''' and submit them in class by October 13, | + | '''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 \ | + | '''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 |
Latest 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:
Problem 1. If and then (as what one would expect by looking at degrees of polynomials) and where and is the multiplication of rationals. (See 090924-2, minute 36:01).
Problem 2. Let be the multiplication operator by the 1-chord diagram , and let be the adjoint of multiplication by on , where is the obvious dual of in . Let be defined by
Verify the following assertions, but submit only your work on assertions 4,5,7,11:
- , where is the identity map and where for any two operators.
- is a degree operator; that is, for all .
- satisfies Leibnitz' law: for any .
- is an algebra morphism: and .
- satisfies the co-Leibnitz law: (why does this deserve the name "the co-Leibnitz law"?).
- is a co-algebra morphism: (where is the co-unit of ) and .
- and hence , where is the ideal generated by in the algebra .
- If is defined by
then for all . - .
- descends to a Hopf algebra morphism , and if is the obvious projection, then is the identity of . (Recall that ).
- .
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.