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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:

  1. , where is the identity map and where for any two operators.
  2. is a degree operator; that is, for all .
  3. satisfies Leibnitz' law: for any .
  4. is an algebra morphism: and .
  5. satisfies the co-Leibnitz law: (why does this deserve the name "the co-Leibnitz law"?).
  6. is a co-algebra morphism: (where is the co-unit of ) and .
  7. and hence , where is the ideal generated by in the algebra .
  8. If is defined by
    then for all .
  9. .
  10. descends to a Hopf algebra morphism , and if is the obvious projection, then is the identity of . (Recall that ).
  11. .

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.