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

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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>.
# <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 15:47, 22 January 2007

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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].

The Equations

The Equations in Functional Terms

A Solution

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