Difference between revisions of "07-1352/Class Notes for January 23"

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===The Algebra===
 
===The Algebra===
  
Let <math>A_n={\mathbb Q}S_n\otimes{\mathbb Q}\langle x_1\ldots x_n\rangle</math> be the vector-space tensor product of the group ring <math>{\mathbb Q}S_n</math> of the permutation group <math>S_n</math> with the completed free associative algebra <math>{\mathbb Q}\langle x_1\ldots x_n\rangle</math> on (non-commuting) generators <math>x_1\ldots x_n</math> (that is, <math>{\mathbb Q}\langle x_1\ldots x_n\rangle</math> is the ring of non-commutative power series in the variables <math>x_1\ldots x_n</math>). We put an algebra structure on <math>A_n</math> as follows:
+
Let <math>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>{\mathbb Q}S_n</math> of the permutation group <math>S_n</math> (with coefficients in <math>{\mathbb Q}[x]</math>, polynomials in the variable <math>x</math>) with the free associative algebra <math>{\mathbb Q}\langle t_1\ldots t_n\rangle</math> on (non-commuting) generators <math>t_1\ldots t_n</math> (that is, <math>{\mathbb Q}\langle t_1\ldots t_n\rangle</math> is the ring of non-commutative polynomials in the variables <math>t_1\ldots t_n</math>). We put an algebra structure on <math>A_n</math> as follows:
 +
 
 +
Let <math>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>S_n</math> on <math>\{1,\ldots,n\}</math> and by formal variables <math>x</math> and <math>t_1\ldots t_n</math>, and let <math>A^1_n</math> be the quotient of <math>A^0_n</math> by the following relations:
 +
# <math>x</math> commutes with everything else.
 +
# The product of permutations is as in the symmetric group <math>S_n</math>.
 +
# If <math>\sigma</math> is a permutation then <math>t_i\sigma=\sigma t_{\sigma i}</math>.
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# <math>[t_i,t_j]=x\sigma_{ij}(t_j-t_i)</math>, where <math>\sigma_{ij}</math> is the transposition of <math>i</math> and <math>j</math>.
 +
Finally, declare that <math>\deg x=\deg t_i=1</math> while <math>\deg\sigma=0</math> for every <math>1\leq i\leq n</math> and every <math>\sigma\in S_n</math>, and let <math>A_n</math> be the graded completion of <math>A^1_n</math>.
  
 
===The Equations===
 
===The Equations===

Revision as of 16:47, 22 January 2007

In Preparation

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

Contents

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^0_n=\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^1_n be the quotient of A^0_n 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^1_n.

The Equations

The Equations in Functional Terms

A Solution