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 $A_{n}={\mathbb {Q} }S_{n}[x]\otimes {\mathbb {Q} }\langle t_{1}\ldots t_{n}\rangle$ be the vector-space tensor product of the group ring ${\mathbb {Q} }S_{n}$ of the permutation group $S_{n}$ (with coefficients in ${\mathbb {Q} }[x]$, polynomials in the variable $x$) with the free associative algebra ${\mathbb {Q} }\langle t_{1}\ldots t_{n}\rangle$ on (non-commuting) generators $t_{1}\ldots t_{n}$ (that is, ${\mathbb {Q} }\langle t_{1}\ldots t_{n}\rangle$ is the ring of non-commutative polynomials in the variables $t_{1}\ldots t_{n}$). We put an algebra structure on $A_{n}$ as follows:

Let $A_{n}^{0}=\langle S_{n},x,t_{1},\ldots t_{n}\rangle$ be the free associative (but non-commutative) algebra generated by the elements of the symmetric group $S_{n}$ on $\{1,\ldots ,n\}$ and by formal variables $x$ and $t_{1}\ldots t_{n}$, and let $A_{n}^{1}$ be the quotient of $A_{n}^{0}$ by the following relations:

1. $x$ commutes with everything else.
2. The product of permutations is as in the symmetric group $S_{n}$.
3. If $\sigma$ is a permutation then $t_{i}\sigma =\sigma t_{\sigma i}$.
4. $[t_{i},t_{j}]=x\sigma _{ij}(t_{j}-t_{i})$, where $\sigma _{ij}$ is the transposition of $i$ and $j$.

Finally, declare that $\deg x=\deg t_{i}=1$ while $\deg \sigma =0$ for every $1\leq i\leq n$ and every $\sigma \in S_{n}$, and let $A_{n}$ be the graded completion of $A_{n}^{1}$.