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

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(The Algebra)
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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>.
 
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>.
  
We say that an element of <math>A_n</math> is "sorted" if it is written in the form <math>x^kt_1^{k_1}t_2^{k_2}\cdots t_n^{k_n}\cdot\sigma</math> where <math>\sigma</math> is a permutation and <math>k</math> and the <math>k_i</math>'s are all non-negative integer. The HOMFLY relations imply that every element of <math>A_n</math> is a linear combinations of sorted elements. Thus as a vector space, <math>A_n</math> can be identified with the ring <math>B_n</math> of power series in the variables <math>x,t_1,\ldots,t_n</math> tensored with the group ring of <math>S_n</math>. The product of <math>A_n</math> is of course very different than that of <math>B_n</math>.
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We say that an element of <math>A_n</math> is "sorted" if it is written in the form <math>x^k\cdot\sigmat_1^{k_1}t_2^{k_2}\cdots t_n^{k_n}</math> where <math>\sigma</math> is a permutation and <math>k</math> and the <math>k_i</math>'s are all non-negative integer. The HOMFLY relations imply that every element of <math>A_n</math> is a linear combinations of sorted elements. Thus as a vector space, <math>A_n</math> can be identified with the ring <math>B_n</math> of power series in the variables <math>x,t_1,\ldots,t_n</math> tensored with the group ring of <math>S_n</math>. The product of <math>A_n</math> is of course very different than that of <math>B_n</math>.
  
 
'''Examples.'''
 
'''Examples.'''
# The general element of <math>A_1</math> is <math>f(x,t_1)(1)</math> where <math>(1)</math> denotes the identity permutation and <math>f(x,t_1)</math> is a power series in two variables <math>x</math> and <math>t_1</math>. <math>A_1</math> is commutative.
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# The general element of <math>A_1</math> is <math>(1)f(x,t_1)</math> where <math>(1)</math> denotes the identity permutation and <math>f(x,t_1)</math> is a power series in two variables <math>x</math> and <math>t_1</math>. <math>A_1</math> is commutative.
# The general element of <math>A_2</math> is <math>f(x,t_1,t_2)(12)+g(x,t_1,t_2)(21)</math> where <math>f</math> and <math>g</math> are power series in three variables and <math>(12)</math> and <math>(21)</math> are the two elements of <math>S_2</math>. <math>A_2</math> is not commutative and its product is non-trivial to describe.
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# The general element of <math>A_2</math> is <math>(12)f(x,t_1,t_2)+(21)g(x,t_1,t_2)</math> where <math>f</math> and <math>g</math> are power series in three variables and <math>(12)</math> and <math>(21)</math> are the two elements of <math>S_2</math>. <math>A_2</math> is not commutative and its product is non-trivial to describe.
 
# The general element of <math>A_3</math> is described using <math>3!=6</math> power series in 4 variables. The general element of <math>A_n</math> is described using n! power series in <math>n+1</math> variables.
 
# The general element of <math>A_3</math> is described using <math>3!=6</math> power series in 4 variables. The general element of <math>A_n</math> is described using n! power series in <math>n+1</math> variables.
  

Revision as of 19:43, 22 January 2007

In Preparation

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

Contents

A HOMFLY Braidor

The Algebra

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 "HOMFLY" 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.

We say that an element of A_n is "sorted" if it is written in the form Failed to parse (unknown function\sigmat): x^k\cdot\sigmat_1^{k_1}t_2^{k_2}\cdots t_n^{k_n}

where \sigma is a permutation and k and the k_i's are all non-negative integer. The HOMFLY relations imply that every element of A_n is a linear combinations of sorted elements. Thus as a vector space, A_n can be identified with the ring B_n of power series in the variables x,t_1,\ldots,t_n tensored with the group ring of S_n. The product of A_n is of course very different than that of B_n.

Examples.

  1. The general element of A_1 is (1)f(x,t_1) where (1) denotes the identity permutation and f(x,t_1) is a power series in two variables x and t_1. A_1 is commutative.
  2. The general element of A_2 is (12)f(x,t_1,t_2)+(21)g(x,t_1,t_2) where f and g are power series in three variables and (12) and (21) are the two elements of S_2. A_2 is not commutative and its product is non-trivial to describe.
  3. The general element of A_3 is described using 3!=6 power series in 4 variables. The general element of A_n is described using n! power series in n+1 variables.

The algebra A_n embeds in A_{n+1} in a trivial way by regarding \{1,\ldots,n\} as a subset of \{1,\ldots,n+1\} in the obvious manner; thus when given an element of A_n we are free to think of it also as an element of A_{n+1}. There is also a non-trivial map \Delta:A_n\to A_{n+1} defined as follows:

  1. \Delta(x)=x.
  2. \Delta(t_i)=t_{i+1}+x\sigma_{1,i+1}.
  3. \Delta acts on permutations by "shifting them one unit to the right", i.e., by identifying \{1,\ldots,n\} with \{2,\ldots,n+1\}\subset\{1,\ldots,n+1\}.

The Equations

We seek to find a "braidor"; an element B of A_2 satisfying:

  • B=(21)+x(12)+(higher order terms).
  • B(\Delta B)B=(\Delta B)B(\Delta B) in A_3.

With the vector space identification of A_n with B_n in mind, we are seeking two power series of three variables each, whose low order behaviour is specified and which are required to satisfy 6 functional equations written in terms of 4 variables.

The Equations in Functional Form

A Solution