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# 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. April 12, 2014

## Transcript

1. Visualizing diagram factorizations in
Temperley–Lieb algebras
Sarah Salmon
Northern Arizona University
Department of Mathematics and Statistics
[email protected]
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

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

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

4. Historical context
• 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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