Visualizing diagram factorizations in Temperley–Lieb algebras

The Temperley-Lieb Algebra of type A, invented by Temperley and Lieb in 1971, is a finite dimensional associative algebra that arose in the context of statistical mechanics. Later in 1971, R. Penrose showed that this algebra can be realized in terms of certain diagrams. Then in 1987, V. Jones showed that this algebra occurs naturally as a quotient of an algebra whose underlying structure is the symmetric group. Similarly, there is also a diagrammatic representation for the Temperley-Lieb algebra of type B involving decorated diagrams. Multiplying diagrams is easy to do. However, taking a given diagram and finding the corresponding reduced factorization is generally difficult. We have an efficient (and colorful) algorithm for obtaining a reduced factorization for Temperley-Lieb diagrams of types A and B.

This talk was given by my undergraduate research student Sarah Salmon (Northern Arizona University) on April 12, 2014 at the 2014 Arizona–Nevada Academy of Science Annual Meeting. This joint work with Michael Hastings.

Transcript

1. Visualizing diagram factorizations in
   Temperley–Lieb algebras
   Sarah Salmon
   Northern Arizona University
   Department of Mathematics and Statistics
   sks254@nau.edu
   Arizona–Nevada Academy of Science
   Annual Meeting
   April 12, 2014
   Directed by Dr. Dana C. Ernst
   Joint work with Michael Hastings
   S. Salmon Visualizing diagram factorizations in TL algebras 1 / 20
   

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2. Type A Temperley–Lieb diagrams
   A diagram of type An
   must satisfy the following requirements:
   • The diagram starts with a box with n + 1 nodes along the north face and n + 1
   nodes along the south face.
   • Every node must be connected to exactly one other node by a single edge.
   • The edges cannot cross.
   • The edges cannot leave the box.
   Example
   Here is a 6-diagram. This is not a diagram.
   S. Salmon Visualizing diagram factorizations in TL algebras 2 / 20
   

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3. The type A Temperley–Lieb diagram algebra
   TL(An
   ) is the Z[δ]-algebra having (n + 1)-diagrams as a basis. When multiplying
   diagrams, it is possible to obtain a loop. In this case, we replace each loop with a
   coefficient δ.
   = δ2
   S. Salmon Visualizing diagram factorizations in TL algebras 3 / 20
   

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4. Historical context
   Comments
   • TL(An
   ) was discovered in 1971 by Temperley and Lieb as an algebra with
   abstract generators and a presentation with the relations above.
   • It first arose in the context of integrable Potts models in statistical mechanics.
   • As well as having applications in physics, TL(An
   ) appears in the framework of
   knot theory, braid groups, Coxeter groups and their corresponding Hecke
   algebras, and subfactors of von Neumann algebras.
   • Penrose/Kauffman used a diagram algebra to model TL(An
   ) in 1971.
   • In 1987, Vaughan Jones (awarded Fields Medal in 1990) recognized that TL(An
   )
   is isomorphic to a particular quotient of the Hecke algebra of type An
   (the
   Coxeter group of type An
   is the symmetric group, Sn+1
   ).
   S. Salmon Visualizing diagram factorizations in TL algebras 4 / 20
   

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5. Type A diagrams
   Example
   For TL(A3
   ), we have the following set of 4-diagrams as a basis:
   S. Salmon Visualizing diagram factorizations in TL algebras 5 / 20
   

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6. Type A diagrams
   Example
   For TL(A3
   ), we have the following set of 4-diagrams as a basis:
   S. Salmon Visualizing diagram factorizations in TL algebras 5 / 20
   

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7. Type A generators and relations
   For TL(An
   ), we define n simple diagrams as follows:
   di
   =
   1 n + 1
   · · · · · ·
   i i + 1
   S. Salmon Visualizing diagram factorizations in TL algebras 6 / 20
   

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8. Type A generators and relations
   For TL(An
   ), we define n simple diagrams as follows:
   di
   =
   1 n + 1
   · · · · · ·
   i i + 1
   Theorem
   TL(An
   ) satisfies the following:
   S. Salmon Visualizing diagram factorizations in TL algebras 6 / 20
   

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9. Type A generators and relations
   For TL(An
   ), we define n simple diagrams as follows:
   di
   =
   1 n + 1
   · · · · · ·
   i i + 1
   Theorem
   TL(An
   ) satisfies the following:
   • d2
   i
   = δdi
   ;
   • di
   dj
   = dj
   di
   when |i − j| > 1;
   • di
   dj
   di
   = di
   when |i − j| = 1.
   S. Salmon Visualizing diagram factorizations in TL algebras 6 / 20
   

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10. Type A generators and relations
    For TL(An
    ), we define n simple diagrams as follows:
    di
    =
    1 n + 1
    · · · · · ·
    i i + 1
    Theorem
    TL(An
    ) satisfies the following:
    • d2
    i
    = δdi
    ;
    • di
    dj
    = dj
    di
    when |i − j| > 1;
    • di
    dj
    di
    = di
    when |i − j| = 1.
    Theorem
    The set of simple diagrams generate all diagrams in the Temperley–Lieb algebra of
    type An
    .
    S. Salmon Visualizing diagram factorizations in TL algebras 6 / 20
    

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11. Products of simple diagrams
    Example
    Consider the product d1
    d2
    d1
    d3
    d2
    d4
    d3
    in TL(A4
    ).
    = =
    Note that d1
    d2
    d1
    d3
    d2
    d4
    d3
    = d1
    d3
    d2
    d4
    d3.
    S. Salmon Visualizing diagram factorizations in TL algebras 7 / 20
    

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12. Products of simple diagrams
    Example
    Consider the product d1
    d2
    d1
    d3
    d2
    d4
    d3
    in TL(A4
    ).
    = =
    Note that d1
    d2
    d1
    d3
    d2
    d4
    d3
    = d1
    d3
    d2
    d4
    d3.
    S. Salmon Visualizing diagram factorizations in TL algebras 7 / 20
    

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13. Factorization of type A
    We have discovered an algorithm to determine the factorization given a diagram.
    S. Salmon Visualizing diagram factorizations in TL algebras 8 / 20
    

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14. Factorization of type A
    We have discovered an algorithm to determine the factorization given a diagram.
    ←→
    S. Salmon Visualizing diagram factorizations in TL algebras 8 / 20
    

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15. Factorization of type A
    We have discovered an algorithm to determine the factorization given a diagram.
    ←→
    S. Salmon Visualizing diagram factorizations in TL algebras 8 / 20
    

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16. Factorization of type A
    We have discovered an algorithm to determine the factorization given a diagram.
    ←→
    S. Salmon Visualizing diagram factorizations in TL algebras 8 / 20
    

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17. Factorization of type A
    We have discovered an algorithm to determine the factorization given a diagram.
    ←→
    S. Salmon Visualizing diagram factorizations in TL algebras 8 / 20
    

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18. Factorization of type A
    We have discovered an algorithm to determine the factorization given a diagram.
    ←→
    S. Salmon Visualizing diagram factorizations in TL algebras 8 / 20
    

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19. Factorization of type A
    We have discovered an algorithm to determine the factorization given a diagram.
    ←→
    By our algorithm, the diagram equals d4
    d7
    d3
    d5
    d8
    d2
    d6
    d1
    d7
    .
    S. Salmon Visualizing diagram factorizations in TL algebras 8 / 20
    

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20. Factorization of type A
    Let’s verify our calculation.
    d4
    d7
    d3
    d5
    d8
    d2
    d6
    d1
    d7
    =
    =
    S. Salmon Visualizing diagram factorizations in TL algebras 9 / 20
    

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21. Type B Temperley–Lieb diagrams
    A diagram must satisfy the restrictions for type A and the following:
    • All decorations must be exposed to the west face;
    • There are a few technical restrictions on what decorations can occur where;
    • All loops (decorated or not) are replaced with a coefficient, δ.
    S. Salmon Visualizing diagram factorizations in TL algebras 10 / 20
    

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22. Type B Temperley–Lieb diagrams
    In TL(Bn
    ), we multiply diagrams as in type A subject to the following relations:
    = = = 2 = 2
    = 2
    S. Salmon Visualizing diagram factorizations in TL algebras 11 / 20
    

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23. Type B diagrams
    Example
    For TL(B3
    ), we have the following set of 4-diagrams as a basis:
    S. Salmon Visualizing diagram factorizations in TL algebras 12 / 20
    

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24. Type B diagrams
    Example
    For TL(B3
    ), we have the following set of 4-diagrams as a basis:
    S. Salmon Visualizing diagram factorizations in TL algebras 12 / 20
    

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25. Type B generators and relations
    For TL(Bn
    ), we define n simple diagrams as follows (i ≥ 2):
    d1
    = · · ·
    1 2 n n + 1
    , di
    =
    1 n + 1
    · · · · · ·
    i i + 1
    S. Salmon Visualizing diagram factorizations in TL algebras 13 / 20
    

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26. Type B generators and relations
    For TL(Bn
    ), we define n simple diagrams as follows (i ≥ 2):
    d1
    = · · ·
    1 2 n n + 1
    , di
    =
    1 n + 1
    · · · · · ·
    i i + 1
    Theorem
    TL(Bn
    ) satisfies the following:
    S. Salmon Visualizing diagram factorizations in TL algebras 13 / 20
    

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27. Type B generators and relations
    For TL(Bn
    ), we define n simple diagrams as follows (i ≥ 2):
    d1
    = · · ·
    1 2 n n + 1
    , di
    =
    1 n + 1
    · · · · · ·
    i i + 1
    Theorem
    TL(Bn
    ) satisfies the following:
    • d2
    i
    = δdi
    ;
    • di
    dj
    = dj
    di
    when |i − j| > 1;
    • di
    dj
    di
    = di
    when |i − j| = 1 and i, j = 1;
    S. Salmon Visualizing diagram factorizations in TL algebras 13 / 20
    

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28. Type B generators and relations
    For TL(Bn
    ), we define n simple diagrams as follows (i ≥ 2):
    d1
    = · · ·
    1 2 n n + 1
    , di
    =
    1 n + 1
    · · · · · ·
    i i + 1
    Theorem
    TL(Bn
    ) satisfies the following:
    • d2
    i
    = δdi
    ;
    • di
    dj
    = dj
    di
    when |i − j| > 1;
    • di
    dj
    di
    = di
    when |i − j| = 1 and i, j = 1;
    • di
    dj
    di
    dj
    = 2di
    dj
    if {i, j} = {1, 2}.
    S. Salmon Visualizing diagram factorizations in TL algebras 13 / 20
    

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29. Type B generators and relations
    For TL(Bn
    ), we define n simple diagrams as follows (i ≥ 2):
    d1
    = · · ·
    1 2 n n + 1
    , di
    =
    1 n + 1
    · · · · · ·
    i i + 1
    Theorem
    TL(Bn
    ) satisfies the following:
    • d2
    i
    = δdi
    ;
    • di
    dj
    = dj
    di
    when |i − j| > 1;
    • di
    dj
    di
    = di
    when |i − j| = 1 and i, j = 1;
    • di
    dj
    di
    dj
    = 2di
    dj
    if {i, j} = {1, 2}.
    Theorem
    The set of simple diagrams generate all diagrams in the Temperley–Lieb algebra of
    type Bn
    .
    S. Salmon Visualizing diagram factorizations in TL algebras 13 / 20
    

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30. Proof of last relation
    Proof
    For i = 1 and j = 2,
    d1
    d2
    d1
    d2
    =
    · · ·
    · · ·
    · · ·
    · · ·
    S. Salmon Visualizing diagram factorizations in TL algebras 14 / 20
    

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31. Proof of last relation
    Proof
    For i = 1 and j = 2,
    d1
    d2
    d1
    d2
    =
    · · ·
    · · ·
    · · ·
    · · ·
    = 2 · · ·
    S. Salmon Visualizing diagram factorizations in TL algebras 14 / 20
    

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32. Proof of last relation
    Proof
    For i = 1 and j = 2,
    d1
    d2
    d1
    d2
    =
    · · ·
    · · ·
    · · ·
    · · ·
    = 2 · · ·
    = 2
    · · ·
    · · ·
    S. Salmon Visualizing diagram factorizations in TL algebras 14 / 20
    

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33. Proof of last relation
    Proof
    For i = 1 and j = 2,
    d1
    d2
    d1
    d2
    =
    · · ·
    · · ·
    · · ·
    · · ·
    = 2 · · ·
    = 2
    · · ·
    · · ·
    = 2d1
    d2
    S. Salmon Visualizing diagram factorizations in TL algebras 14 / 20
    

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34. Product of simple diagrams in type B
    Example
    Here is an example of a product of several simple diagrams in TL(B4
    ).
    d1
    d2
    d4
    d1
    d3
    d2
    =
    =
    S. Salmon Visualizing diagram factorizations in TL algebras 15 / 20
    

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35. Factorization of type B
    Example
    Using a diagram, let’s work towards the factorization.
    S. Salmon Visualizing diagram factorizations in TL algebras 16 / 20
    

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36. Factorization of type B
    Example
    Using a diagram, let’s work towards the factorization.
    ←→
    S. Salmon Visualizing diagram factorizations in TL algebras 16 / 20
    

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37. Factorization of type B
    Example
    Using a diagram, let’s work towards the factorization.
    ←→
    S. Salmon Visualizing diagram factorizations in TL algebras 16 / 20
    

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38. Factorization of type B
    Example
    Using a diagram, let’s work towards the factorization.
    ←→
    S. Salmon Visualizing diagram factorizations in TL algebras 16 / 20
    

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39. Factorization of type B
    Example
    Using a diagram, let’s work towards the factorization.
    ←→
    S. Salmon Visualizing diagram factorizations in TL algebras 16 / 20
    

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40. Factorization of type B
    Example
    Using a diagram, let’s work towards the factorization.
    ←→
    S. Salmon Visualizing diagram factorizations in TL algebras 16 / 20
    

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41. Factorization of type B
    Example
    Using a diagram, let’s work towards the factorization.
    ←→
    S. Salmon Visualizing diagram factorizations in TL algebras 16 / 20
    

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42. Factorization of type B
    Example
    Using a diagram, let’s work towards the factorization.
    ←→
    S. Salmon Visualizing diagram factorizations in TL algebras 16 / 20
    

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43. Factorization of type B
    Example
    Using a diagram, let’s work towards the factorization.
    ←→
    S. Salmon Visualizing diagram factorizations in TL algebras 16 / 20
    

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44. Factorization of type B
    Example
    Using a diagram, let’s work towards the factorization.
    ←→
    S. Salmon Visualizing diagram factorizations in TL algebras 16 / 20
    

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45. Factorization of type B
    Example
    Using a diagram, let’s work towards the factorization.
    ←→
    Therefore, the original diagram equals
    d1
    d4
    d8
    d10
    d3
    d5
    d9
    d2
    d4
    d6
    d1
    d3
    d5
    d7
    d2
    d4
    d6
    d8
    d1
    d3
    d5
    d2
    d4
    .
    S. Salmon Visualizing diagram factorizations in TL algebras 16 / 20
    

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46. Factorization of type B
    Let’s check our calculation:
    d1
    d4
    d8
    d10
    d3
    d5
    d9
    d2
    d4
    d6
    d1
    d3
    d5
    d7
    d2
    d4
    d6
    d8
    d1
    d3
    d5
    d2
    d4
    = =
    S. Salmon Visualizing diagram factorizations in TL algebras 17 / 20
    

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47. An exception
    Example
    There is one case where we must slightly adjust our algorithm.
    S. Salmon Visualizing diagram factorizations in TL algebras 18 / 20
    

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48. An exception
    Example
    There is one case where we must slightly adjust our algorithm.
    ←→
    S. Salmon Visualizing diagram factorizations in TL algebras 18 / 20
    

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49. An exception
    Example
    There is one case where we must slightly adjust our algorithm.
    ←→
    S. Salmon Visualizing diagram factorizations in TL algebras 18 / 20
    

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50. An exception
    Example
    There is one case where we must slightly adjust our algorithm.
    ←→
    S. Salmon Visualizing diagram factorizations in TL algebras 18 / 20
    

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51. An exception
    Example
    There is one case where we must slightly adjust our algorithm.
    ←→
    S. Salmon Visualizing diagram factorizations in TL algebras 18 / 20
    

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52. An exception
    Example
    There is one case where we must slightly adjust our algorithm.
    ←→
    S. Salmon Visualizing diagram factorizations in TL algebras 18 / 20
    

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53. An exception
    Example
    There is one case where we must slightly adjust our algorithm.
    ←→
    S. Salmon Visualizing diagram factorizations in TL algebras 18 / 20
    

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54. An exception
    Example
    There is one case where we must slightly adjust our algorithm.
    ←→
    S. Salmon Visualizing diagram factorizations in TL algebras 18 / 20
    

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55. An exception
    Example
    There is one case where we must slightly adjust our algorithm.
    ←→
    S. Salmon Visualizing diagram factorizations in TL algebras 18 / 20
    

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56. An exception
    Example
    There is one case where we must slightly adjust our algorithm.
    ←→
    S. Salmon Visualizing diagram factorizations in TL algebras 18 / 20
    

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57. An exception
    Example
    There is one case where we must slightly adjust our algorithm.
    ←→
    This diagram equals d5
    d4
    d6
    d3
    d2
    d1
    d2
    d3
    S. Salmon Visualizing diagram factorizations in TL algebras 18 / 20
    

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58. An exception
    Here is the product of simple diagrams:
    d5
    d4
    d6
    d3
    d2
    d1
    d2
    d3
    =
    =
    S. Salmon Visualizing diagram factorizations in TL algebras 19 / 20
    

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59. An exception
    Here is the product of simple diagrams:
    d5
    d4
    d6
    d3
    d2
    d1
    d2
    d3
    =
    =
    S. Salmon Visualizing diagram factorizations in TL algebras 19 / 20
    

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60. Further work and acknowledgements
    Further work
    Will our algorithm work on other types where diagrammatic representations are
    known to exist? For example, TL(Cn
    ):
    Acknowledgements
    • Northern Arizona University
    • Department of Mathematics and Statistics
    • Dr. Dana Ernst
    S. Salmon Visualizing diagram factorizations in TL algebras 20 / 20
    

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