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

Problem 1. Let {\mathfrak g}_1 and {\mathfrak g}_2 be finite dimensional metrized Lie algebras, let {\mathfrak g}_1\oplus{\mathfrak g}_2 denote their direct sum with the obvious "orthogonal" bracket and metric, and let m be the canonical isomorphism m:{\mathcal U}({\mathfrak g}_1)\otimes{\mathcal U}({\mathfrak g}_2)\to{\mathcal U}({\mathfrak g}_1\oplus{\mathfrak g}_2). Prove that

{\mathcal T}_{{\mathfrak g}_1\oplus{\mathfrak g}_2} = m\circ({\mathcal T}_{{\mathfrak g}_1}\otimes{\mathcal T}_{{\mathfrak g}_2})\circ\Box,

where \Box:{\mathcal A}(\uparrow)\to{\mathcal A}(\uparrow)\otimes{\mathcal A}(\uparrow) is the co-product and {\mathcal T}_{{\mathfrak g}} denotes the {\mathcal U}({\mathfrak g})-valued "tensor map" on {\mathcal A}. Can you relate this with the first problem of HW1?

Problem 2.

  1. Find a concise algorithm to compute the weight system W_{so} associated with the Lie algebra so(N) in its defining representation.
  2. Verify that your algorithm indeed satisfies the 4T relation.

Problem 3. The Kauffman polynomial F(K)(a,z) (see [Kauffman]) of a knot or link K is a^{-w(K)}L(K) where w(L) is the writhe of K and where L(K) is the regular isotopy invariant defined by the skein relations

L(s_\pm)=a^{\pm 1}L(s))

(here s is a strand and s_\pm is the same strand with a \pm kink added) and

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

and by the initial condition L(\bigcirc)=1. State and prove the relationship between F and W_{so}.

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.