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Solve the following problems and submit them in class by November 3, 2009:

Problem 1. Let and be finite dimensional metrized Lie algebras, let denote their direct sum with the obvious "orthogonal" bracket and metric, and let be the canonical isomorphism . Prove that


where is the co-product and denotes the -valued "tensor map" on . Can you relate this with the first problem of HW1?

Problem 2.

  1. Find a concise algorithm to compute the weight system associated with the Lie algebra in its defining representation.
  2. Verify that your algorithm indeed satisfies the relation.

Problem 3. The Kauffman polynomial (see [Kauffman]) of a knot or link is where is the writhe of and where is the regular isotopy invariant defined by the skein relations

(here is a strand and is the same strand with a kink added) and

Failed to parse (unknown function "\backoverslash"): {\displaystyle L(\backoverslash)+L(\slashoverback) = z\left(L(\smoothing)+L(\hsmoothing)\right)}

and by the initial condition . State and prove the relationship between and .

Mandatory but unenforced. Find yourself in the class photo and identify yourself as explained in the photo page.


[Kauffman] ^  L. H. Kauffman, An invariant of regular isotopy, Trans. Amer. Math. Soc. 312 (1990) 417-471.