07-1352/Class Notes for January 23

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In Preparation

The information below is preliminary and cannot be trusted! (v)

A HOMFLY Braidor

The Algebra

Let [math]\displaystyle{ A_n={\mathbb Q}S_n[x]\otimes{\mathbb Q}\langle t_1\ldots t_n\rangle }[/math] be the vector-space tensor product of the group ring [math]\displaystyle{ {\mathbb Q}S_n }[/math] of the permutation group [math]\displaystyle{ S_n }[/math] (with coefficients in [math]\displaystyle{ {\mathbb Q}[x] }[/math], polynomials in the variable [math]\displaystyle{ x }[/math]) with the free associative algebra [math]\displaystyle{ {\mathbb Q}\langle t_1\ldots t_n\rangle }[/math] on (non-commuting) generators [math]\displaystyle{ t_1\ldots t_n }[/math] (that is, [math]\displaystyle{ {\mathbb Q}\langle t_1\ldots t_n\rangle }[/math] is the ring of non-commutative polynomials in the variables [math]\displaystyle{ t_1\ldots t_n }[/math]). We put an algebra structure on [math]\displaystyle{ A_n }[/math] as follows:

Let [math]\displaystyle{ A^0_n=\langle S_n, x, t_1,\ldots t_n\rangle }[/math] be the free associative (but non-commutative) algebra generated by the elements of the symmetric group [math]\displaystyle{ S_n }[/math] on [math]\displaystyle{ \{1,\ldots,n\} }[/math] and by formal variables [math]\displaystyle{ x }[/math] and [math]\displaystyle{ t_1\ldots t_n }[/math], and let [math]\displaystyle{ A^1_n }[/math] be the quotient of [math]\displaystyle{ A^0_n }[/math] by the following relations:

1. [math]\displaystyle{ x }[/math] commutes with everything else.
2. The product of permutations is as in the symmetric group [math]\displaystyle{ S_n }[/math].
3. If [math]\displaystyle{ \sigma }[/math] is a permutation then [math]\displaystyle{ t_i\sigma=\sigma t_{\sigma i} }[/math].
4. [math]\displaystyle{ [t_i,t_j]=x\sigma_{ij}(t_j-t_i) }[/math], where [math]\displaystyle{ \sigma_{ij} }[/math] is the transposition of [math]\displaystyle{ i }[/math] and [math]\displaystyle{ j }[/math].

Finally, declare that [math]\displaystyle{ \deg x=\deg t_i=1 }[/math] while [math]\displaystyle{ \deg\sigma=0 }[/math] for every [math]\displaystyle{ 1\leq i\leq n }[/math] and every [math]\displaystyle{ \sigma\in S_n }[/math], and let [math]\displaystyle{ A_n }[/math] be the graded completion of [math]\displaystyle{ A^1_n }[/math].