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: