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 $A_n={\mathbb Q}S_n\otimes{\mathbb Q}\langle x_1\ldots x_n\rangle$ be the vector-space tensor product of the group ring ${\mathbb Q}S_n$ of the permutation group $S_n$ with the completed free associative algebra ${\mathbb Q}\langle x_1\ldots x_n\rangle$ on (non-commuting) generators $x_1\ldots x_n$ (that is, ${\mathbb Q}\langle x_1\ldots x_n\rangle$ is the ring of non-commutative power series in the variables $x_1\ldots x_n$). We put an algebra structure on $A_n$ as follows: