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| In [[statistical mechanics]], the '''Temperley–Lieb algebra''' is an algebra from which are built certain [[transfer matrix|transfer matrices]], invented by [[Harold Neville Vazeille Temperley|Neville Temperley]] and [[Elliott H. Lieb|Elliott Lieb]]. It is also related to [[integrable model]]s, [[knot theory]] and the [[braid group]], [[quantum groups]] and [[subfactor]]s of [[von Neumann algebra]]s.
| | Emilia Shryock is my title but you can call me anything you like. For a whilst I've been in South Dakota and my parents live close by. To play baseball is the pastime he will never quit doing. My day job is a meter reader.<br><br>my web page ... [http://www.hotporn123.com/blog/154369 home std test] |
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| ==Definition==
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| Let <math>R</math> be a [[commutative ring]] and fix <math>\delta \in R</math>. The Temperley–Lieb algebra <math>TL_n(\delta)</math> is the [[algebra (ring theory)|<math>R</math>-algebra]] generated by the elements <math>U_1, U_2, \ldots, U_{n-1}</math>, subject to the Jones relations:
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| *<math>U_i^2 = \delta U_i</math> for all <math>1 \leq i \leq n-1</math>
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| *<math>U_i U_{i+1} U_i = U_i</math> for all <math>1 \leq i \leq n-2</math>
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| *<math>U_i U_{i-1} U_i = U_i</math> for all <math>2 \leq i \leq n-1</math>
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| *<math>U_i U_j = U_j U_i</math> for all <math>1 \leq i,j \leq n-1</math> such that <math>|i-j| \neq 1</math>
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| <math>TL_n(\delta)</math> may be represented diagrammatically as the vector space over noncrossing pairings on a rectangle with ''n'' points on two opposite sides. The five basis elements of <math>TL_3(\delta)</math> are the following:
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| [[File:Temperley-lieb (horizontal).svg|340px|Basis of the Temperley–Lieb algebra <math>TL_3(\delta)</math>]].
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| Multiplication on basis elements can be performed by placing two rectangles side by side, and replacing any closed loops by a factor of ''δ'', for example:
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| [[File:Factor-a.svg|50px]] × [[File:Factor-b.svg|50px]] = [[File:Factor-a.svg|50px]][[File:Factor-b.svg|50px]] = δ [[File:Concatenation-ab.svg|50px]].
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| The identity element is the diagram in which each point is connected to the one directly across the rectangle from it, and the generator <math>U_i</math> is the diagram in which the ''i''th point is connected to the ''i+1''th point, the ''2n − i + 1''th point is connected to the ''2n − i''th point, and all other points are connected to the point directly across the rectangle. The generators of <math>TL_5(\delta)</math> are:
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| [[File:Temperley-Lieb (generateurs).svg|340px|Generators of the Temperley–Lieb algebra <math>TL_5(\delta)</math>]]
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| From left ot right, the unit 1 and the generators U<sub>1</sub>, U<sub>2</sub>, U<sub>3</sub>, U<sub>4</sub>.
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| The Jones relations can be seen graphically:
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| [[File:E 2 Temperley.svg|50px]] [[File:E 2 Temperley.svg|50px]] = δ [[File:E 2 Temperley.svg|50px]]
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| [[File:E 2 Temperley.svg|50px]] [[File:E 3 Temperley.svg|50px]] [[File:E 2 Temperley.svg|50px]] = [[File:E 2 Temperley.svg|50px]]
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| [[File:E 1 Temperley.svg|50px]] [[File:E 4 Temperley.svg|50px]] = [[File:E 4 Temperley.svg|50px]] [[File:E 1 Temperley.svg|50px]]
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| ==The Temperley-Lieb Hamiltonian==
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| Consider an interaction-round-a-face model e.g. a square [[Lattice model (physics)|lattice model]] and let <math>L</math> be the number of sites on the lattice. Following Temperley and Lieb<ref>Temperley N. and Lieb E., (1971), Proc. R. Soc. A 322 251.</ref> we define the Temperley-Lieb [[Hamiltonian (quantum mechanics)|hamiltonian]] (the TL hamiltonian) as
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| <math> \mathcal{H} = \sum_{j=1}^{L-1} (1 - e_j) </math>
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| where <math>e_j = U(\lambda)/\sin\lambda</math>, for some spectral parameter <math>\lambda \in R</math>.
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| ===Applications===
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| We will firstly consider the case <math>L = 3</math>. The TL hamiltonian is <math>\mathcal{H} = 2 - e_1 - e_2 </math>, namely
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| <math>\mathcal{H}</math> = 2 [[File:Unit 3 Temperley.svg|50px]] - [[File:E 1 3 Temperley.svg|50px]] - [[File:E 2 3 Temperley.svg|50px]].
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| We have two possible states,
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| [[File:BS1-Temperley-Lieb.svg|40px]] and [[File:BS2-Temperley-Lieb.svg|40px]].
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| In acting by <math>\mathcal{H}</math> on these states, we find
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| <math>\mathcal{H}</math> [[File:BS1-Temperley-Lieb.svg|40px]] = 2 [[File:Unit 3 Temperley.svg|50px]][[File:BS1-Temperley-Lieb.svg|40px]] - [[File:E 1 3 Temperley.svg|50px]][[File:BS1-Temperley-Lieb.svg|40px]] - [[File:E 2 3 Temperley.svg|50px]][[File:BS1-Temperley-Lieb.svg|40px]] = [[File:BS1-Temperley-Lieb.svg|40px]] - [[File:BS2-Temperley-Lieb.svg|40px]],
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| and
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| <math>\mathcal{H}</math> [[File:BS2-Temperley-Lieb.svg|40px]] = 2 [[File:Unit 3 Temperley.svg|50px]][[File:BS2-Temperley-Lieb.svg|40px]] - [[File:E 1 3 Temperley.svg|50px]][[File:BS2-Temperley-Lieb.svg|40px]] - [[File:E 2 3 Temperley.svg|50px]][[File:BS2-Temperley-Lieb.svg|40px]] = - [[File:BS1-Temperley-Lieb.svg|40px]] + [[File:BS2-Temperley-Lieb.svg|40px]].
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| Writing <math>\mathcal{H}</math> as a matrix in the basis of possible states we have,
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| <math> \mathcal{H} = \left(\begin{array}{rr}
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| 1 & -1\\
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| -1 & 1
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| \end{array}\right)
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| </math>
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| The eigenvector of <math>\mathcal{H}</math> with the ''lowest'' [[Eigenvalues and eigenvectors|eigenvalue]] is known as the [[ground state]]. In this case, the lowest eigenvalue <math>\lambda_0</math> for <math>\mathcal{H}</math> is <math>\lambda_0 = 0</math>. The corresponding [[Eigenvalues and eigenvectors|eigenvector]] is <math>\psi_0 = (1, 1)</math>. As we vary the number of sites <math>L</math> we find the following table<ref name="bach">Batchelor M., de Gier J. and Nienhuis B., (2001), The quantum symmetric <math>XXZ</math> chain at <math>\Delta = -1/2</math>, alternating-sign matrices and plane partitions, J. Phys. A 34, L265-L270.</ref>
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| {| class="wikitable"
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| |-
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| ! <math>L</math>
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| ! <math>\psi_0</math>
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| ! <math>L</math>
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| ! <math>\psi_0</math>
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| |-
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| | 2
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| | (1)
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| |3
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| |(1, 1)
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| |-
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| | 4
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| |(2, 1)
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| |5
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| |<math>(3_3, 1_2)</math>
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| |-
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| | 6
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| | <math>(11, 5_2,4, 1)</math>
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| |7
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| |<math>(26_4, 10_2, 9_2, 8_2, 5_2, 1_2)</math>
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| |-
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| |8
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| |<math>(170, 75_2, 71, 56_2, 50, 30, 14_4, 6, 1)</math>
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| |9
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| |<math>(646, \ldots)</math>
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| |-
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| |<math>\vdots</math>
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| |<math>\vdots</math>
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| |<math>\vdots</math>
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| |<math>\vdots</math>
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| |-
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| |}
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| where we have use the notation <math>m_n = (m, \ldots, m)</math> <math>n</math>-times i.e. <math>5_2 = (5, 5)</math>.
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| ===Combinatorial Properties===
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| An interesting observation is that the largest components of the ground state of <math>\mathcal{H}</math> have a combinatorial enumeration as we vary the number of sites,<ref>de Gier J., (2005), Loops, matchings and alternating-sign matrices, Discrete Mathematics Volume 298, Issues 1-3, Pages 365-388.</ref> as was first observed by [[Murray Batchelor]], Jan de Gier and Bernard Nienhuis.<ref name="bach">Batchelor M., de Gier J. and Nienhuis B., (2001), The quantum symmetric <math>XXZ</math> chain at <math>\Delta = -1/2</math>, alternating-sign matrices and plane partitions, J. Phys. A 34, L265-L270.</ref> Using the resources of the [[on-line encyclopedia of integer sequences]], Batchelor ''et al.'' found, for an even numbers of sites
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| <math>
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| 1, 2, 11, 170, \ldots = \prod_{j=0}^{n-1} \left( 3j + 1\right)\frac{ (2j)!(6j)!}{(4j)!(4j + 1)!}
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| </math>
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| and for an odd numbers of sites
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| <math>
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| 1, 3, 26, 646, \ldots = \prod_{j=0}^{n-1} (3j+2)\frac{ (2j + 2)!(6j + 3)!}{(4j + 2)!(4j + 3)!}.
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| </math>
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| Surprisingly, these sequences corresponded to well known combinatorial objects. For <math>L</math> even, this sequence corresponded to cyclically symmetric transpose complement plane partitions and for <math>L</math> odd these corresponded to <math>(2n+1)\times(2n+1)</math> [[Alternating sign matrix|alternating sign matrices]] symmetric about the vertical axis.
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| ==References==
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| <references/>
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| ==Further reading==
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| *[[Louis H. Kauffman]], [http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V1J-45DHSCR-J&_user=10&_coverDate=12%2F31%2F1987&_rdoc=9&_fmt=high&_orig=browse&_srch=doc-info(%23toc%235676%231987%23999739996%23292694%23FLP%23display%23Volume)&_cdi=5676&_sort=d&_docanchor=&_ct=9&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=cca311a0762fc3d6a7a6f284a10a5c68 ''State Models and the Jones Polynomial''.] [[Topology (journal)|Topology]], 26(3):395-407, 1987.
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| *[[Rodney J. Baxter|R.J. Baxter]], [http://tpsrv.anu.edu.au/Members/baxter/book ''Exactly solved models in statistical mechanics''] Academic Press Inc. (1982)
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| *[[Harold Neville Vazeille Temperley|N. Temperley]], [[Elliott H. Lieb|E. Lieb]], [http://www.jstor.org/stable/77727 ''Relations between the percolation and colouring problem and other graph-theoretical problems associated with regular planar lattices: some exact results for the percolation problem''.] Proceedings of the Royal Society Series A 322 (1971), 251-280.
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| {{DEFAULTSORT:Temperley-Lieb algebra}}
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| [[Category:Von Neumann algebras]]
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| [[Category:Algebra]]
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| [[Category:Knot theory]]
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| [[Category:Braids]]
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| [[Category:Diagram algebras]]
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Emilia Shryock is my title but you can call me anything you like. For a whilst I've been in South Dakota and my parents live close by. To play baseball is the pastime he will never quit doing. My day job is a meter reader.
my web page ... home std test