Junction Type Representations of the Temperley-Lieb Algebra and Associated Symmetries

Inspired by earlier works on representations of the Temperley-Lieb algebra we introduce a novel family of representations of the algebra. This may be seen as a generalization of the so called asymmetric twin representation. The underlying symmetry algebra is also examined and it is shown that in add...

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Дата:	2010
Автори:	Doikou, A., Karaiskos, N.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут математики НАН України 2010
Назва видання:	Symmetry, Integrability and Geometry: Methods and Applications
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/146516
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Цитувати:	Junction Type Representations of the Temperley-Lieb Algebra and Associated Symmetries / A. Doikou, N. Karaiskos // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 24 назв. — англ.

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spelling	irk-123456789-1465162019-02-10T01:25:17Z Junction Type Representations of the Temperley-Lieb Algebra and Associated Symmetries Doikou, A. Karaiskos, N. Inspired by earlier works on representations of the Temperley-Lieb algebra we introduce a novel family of representations of the algebra. This may be seen as a generalization of the so called asymmetric twin representation. The underlying symmetry algebra is also examined and it is shown that in addition to certain obvious exact quantum symmetries non trivial quantum algebraic realizations that exactly commute with the representation also exist. Non trivial representations of the boundary Temperley-Lieb algebra as well as the related residual symmetries are also discussed. The corresponding novel R and K matrices solutions of the Yang-Baxter and reflection equations are identified, the relevant quantum spin chain is also constructed and its exact symmetries are studied. 2010 Article Junction Type Representations of the Temperley-Lieb Algebra and Associated Symmetries / A. Doikou, N. Karaiskos // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 24 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 81R50; 17B37; 17B80 DOI:10.3842/SIGMA.2010.089 http://dspace.nbuv.gov.ua/handle/123456789/146516 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	Inspired by earlier works on representations of the Temperley-Lieb algebra we introduce a novel family of representations of the algebra. This may be seen as a generalization of the so called asymmetric twin representation. The underlying symmetry algebra is also examined and it is shown that in addition to certain obvious exact quantum symmetries non trivial quantum algebraic realizations that exactly commute with the representation also exist. Non trivial representations of the boundary Temperley-Lieb algebra as well as the related residual symmetries are also discussed. The corresponding novel R and K matrices solutions of the Yang-Baxter and reflection equations are identified, the relevant quantum spin chain is also constructed and its exact symmetries are studied.
format	Article
author	Doikou, A. Karaiskos, N.
spellingShingle	Doikou, A. Karaiskos, N. Junction Type Representations of the Temperley-Lieb Algebra and Associated Symmetries Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Doikou, A. Karaiskos, N.
author_sort	Doikou, A.
title	Junction Type Representations of the Temperley-Lieb Algebra and Associated Symmetries
title_short	Junction Type Representations of the Temperley-Lieb Algebra and Associated Symmetries
title_full	Junction Type Representations of the Temperley-Lieb Algebra and Associated Symmetries
title_fullStr	Junction Type Representations of the Temperley-Lieb Algebra and Associated Symmetries
title_full_unstemmed	Junction Type Representations of the Temperley-Lieb Algebra and Associated Symmetries
title_sort	junction type representations of the temperley-lieb algebra and associated symmetries
publisher	Інститут математики НАН України
publishDate	2010
url	http://dspace.nbuv.gov.ua/handle/123456789/146516
citation_txt	Junction Type Representations of the Temperley-Lieb Algebra and Associated Symmetries / A. Doikou, N. Karaiskos // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 24 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT doikoua junctiontyperepresentationsofthetemperleyliebalgebraandassociatedsymmetries AT karaiskosn junctiontyperepresentationsofthetemperleyliebalgebraandassociatedsymmetries
first_indexed	2025-07-11T00:09:58Z
last_indexed	2025-07-11T00:09:58Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 6 (2010), 089, 19 pages Junction Type Representations of the Temperley–Lieb Algebra and Associated Symmetries? Anastasia DOIKOU † and Nikos KARAISKOS †‡ † Department of Engineering Sciences, University of Patras, GR-26500 Patras, Greece E-mail: adoikou@upatras.gr, nkaraiskos@upatras.gr ‡ Centre de Physique Théorique, Ecole Polytechnique, CNRS-UMR 7644, 91128 Palaiseau, France Received September 06, 2010, in final form November 29, 2010; Published online December 07, 2010 doi:10.3842/SIGMA.2010.089 Abstract. Inspired by earlier works on representations of the Temperley–Lieb algebra we introduce a novel family of representations of the algebra. This may be seen as a generaliza- tion of the so called asymmetric twin representation. The underlying symmetry algebra is also examined and it is shown that in addition to certain obvious exact quantum symmetries non trivial quantum algebraic realizations that exactly commute with the representation also exist. Non trivial representations of the boundary Temperley–Lieb algebra as well as the related residual symmetries are also discussed. The corresponding novel R and K matrices solutions of the Yang–Baxter and reflection equations are identified, the relevant quantum spin chain is also constructed and its exact symmetries are studied. Key words: quantum integrability; Temperley–Lieb algebras; symmetries associated to in- tegrable models 2010 Mathematics Subject Classification: 81R50; 17B37; 17B80 1 Introduction There has been an increasing activity in recent years on the study of representations of (affine) Hecke algebras [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21] and their quotients, which can provide solutions [12, 13, 14, 15, 16, 17, 18, 19, 20] of the Yang–Baxter [1] and the reflection equations [4, 23]. Such investigations are of great physical significance due to the fact that the obtained solutions can be implemented into physical integrable systems, and conformal field theories with open boundary conditions that preserve integrability (see e.g. [7, 8, 15, 20, 22]), or lead to the construction of novel physical systems as will be transparent in the present investigation (see also [7]). In the present article, we focus on the investigation of novel representations in the context of the Temperley–Lieb (TL) algebra [24, 18], the so called junction representation. In fact, this may be seen as a natural generalization of the asymmetric twin (or cable) representation studied in [17, 7, 6]. More precisely, the representation is introduced and it is shown that it satisfies the defining relations of the algebra, provided that certain constraints are satisfied. In addition, the existence of non trivial quantum algebras that are exact symmetries of the aforementioned representation is investigated. Our analysis is further extended in the case of the boundary Temperley–Lieb (blob) algebra, and non trivial representations for the “boundary” element of the algebra are identified. It is worth noting that the representation leads naturally to novel physical integrable systems – quantum spin chains, with non-trivial integrable boundary conditions. It is thus clear that the whole analysis presents not only mathematical, but physical interest as well, given that such ?This paper is a contribution to the Proceedings of the International Workshop “Recent Advances in Quantum Integrable Systems”. The full collection is available at http://www.emis.de/journals/SIGMA/RAQIS2010.html mailto:adoikou@upatras.gr mailto:nkaraiskos@upatras.gr http://dx.doi.org/10.3842/SIGMA.2010.089 http://www.emis.de/journals/SIGMA/RAQIS2010.html 2 A. Doikou and N. Karaiskos investigations provide new boundary conditions that may alter the physical behavior as well as the symmetries associated to the corresponding system (see e.g. [7, 8, 6]). The outline of this article is as follows: in the next section the definitions of the Temperley– Lieb algebra are reviewed, the junction representation is introduced, and it is shown that it satisfies the Temperley–Lieb algebra defining relations. The exact symmetries associated to the junction representation are then examined, certain obvious and non-trivial representations of quantum algebras are introduced, and turn out to exactly commute with all the elements of the Temperley–Lieb algebra in the junction representation. Explicit proof for the first non trivial case is provided, and expressions of a whole family of representations of the quantum algebra for the generic case are conjectured. In Section 3 the boundary Temperley–Lieb algebras and the related representations are discussed, and the residual symmetries are examined. In the last section we introduce the relevant quantum spin chain, and we also examine the associa- ted symmetries of the open transfer matrix. In the Appendices various technical points are illustrated. 2 Temperley–Lieb algebra and representations Let us briefly review the basic notions associated to the Temperley–Lieb algebra [24, 18]. The Temperley–Lieb TN (q) algebra is defined by the generators Ui, i = 1, . . . , N satisfying U2 i = − ( q + q−1 ) Ui, UiUi±1Ui = Ui, [Ui, Uj ] = 0, \|i− j\| > 1. (1) A well known representation of the Temperley–Lieb algebra is the XXZ one. More precisely, consider the matrix U = 2∑ a 6=b=1 eab ⊗ eba − 2∑ a 6=b=1 q−sgn(a−b)eaa ⊗ ebb, (2) where the matrices eab are in this case 2 × 2 matrices defined as: (eab)cd = δacδbd. Then take the representation of the Temperley–Lieb algebra ρ : Tn(q) 7→ End((C2)⊗N ) such that ρ(Ui) = I⊗ · · · ⊗ I⊗ U︸︷︷︸ i, i+1 ⊗I⊗ · · · ⊗ I. Inspired by the asymmetric twin (or cable) representation [17, 7] of the Temperley–Lieb algebra we introduce a novel representation, which we shall call for obvious reasons the junction representation. This involves essentially n copies of the XXZ representation i.e. Θ : Tn(q) 7→ End(((C2)⊗n)⊗N ) Θ(Ul) = n∏ i=1 ρai(Ul(i)), (3) where ai is the parameter associated with the i-th copy of the Temperley–Lieb algebra and is a representation of the Temperley–Lieb algebra itself provided that: (−1)n n∏ i=1 ( ai + a−1 i ) = − ( q + q−1 ) . (4) The latter condition apparently follows from the quadratic relation (Hecke condition) of the Temperley–Lieb algebra. Now observe that the matrices ρai(Ul(i)) commute among each other, Junction Type Representations of the Temperley–Lieb Algebra 3 since they act on different vector spaces. Having that in mind and taking into account that each one of the matrices satisfy the TL algebra, one can immediately show that Θ provides a representation of the TL algebra. Constraints are only entailed from the condition (4). One then obtains n equations with n unknown quantities (see Appendix A for more details on the Hecke condition), hence ai’s can be explicitly determined. Recall that the asymmetric twin representation involves two copies of the XXZ representation i.e. n = 2. Note that we consider here the following sequence of spaces in the tensor product: · · ·Vl(1)⊗Vl(2)⊗· · ·⊗Vl(n)⊗V(l+1)(1)⊗ V(l+1)(2) ⊗ · · · ⊗ V(l+1)(n) · · · , V ≡ C2. It is clear that the total number of C2 spaces (i.e. total number of “sites”) is nN . The junction representation can be schematically represented as follows: ... ... ... ... ... ... ... ... ... ... ... ... s s s s s s s s s s s s s s s s s s s s 1(1) 2(1) 3(1) 4(1) · · · N (1) 1(2) 2(2) 3(2) 4(2) · · · N (2) 1(i) 2(i) 3(i) 4(i) · · · N (i) 1(n) 2(n) 3(n) 4(n) · · · N (n) r �BB t t t t t1 2 3 4 · · · N The thick line schematically presents the junction representation, showing somehow the “fu- sion” of n copies of the XXZ representation. In particular, each one of the indices l appea- ring there may be seen as a “fusion” of the indices l(i) as: l ≡ 〈l(1) l(2) . . . l(N)〉, l + 1 ≡ 〈(l + 1)(1) (l + 1)(2) . . . (l + 1)(N)〉 and so on. Let us now explicitly express each one of the terms in the product (3), defining the junction representation: ρam(Ul(m)) = · · · I⊗ ∑ a 6=b I⊗ · · · ⊗ I︸︷︷︸ m−1 ⊗ eab︸︷︷︸ l(m) ⊗ I⊗ · · · ⊗ I︸︷︷︸ n−1 ⊗ eba︸︷︷︸ (l+1)(m) ⊗ I⊗ · · · ⊗ I︸︷︷︸ n−m − ∑ a 6=b a−sgn(a−b) m I⊗ · · · ⊗ I︸︷︷︸ m−1 ⊗ eaa︸︷︷︸ l(m) ⊗ I⊗ · · · ⊗ I︸︷︷︸ n−1 ⊗ ebb︸︷︷︸ (l+1)(m) ⊗ I⊗ · · · ⊗ I︸︷︷︸ n−m ⊗ I · · · with m ∈ {1, . . . , n}. The matrix above clearly acts non-trivially on the spaces C2 l(m) ⊗ C2 (l+1)(m) . 2.1 The symmetry: n = 3 For the sake of simplicity and for illustrating purposes, we shall focus in this section on the simplest non-trivial case where n = 3. Nevertheless, expressions on the symmetries associated to higher n will be presented in the next subsection. 4 A. Doikou and N. Karaiskos The junction representation of the Temperley–Lieb algebra is in this case Θ : TN (q) 7→ End((C2 ⊗ C2 ⊗ C2)⊗N ) Θ(Ul) = ρa1(Ul(1)) ρa2(Ul(2)) ρa3(Ul(3)). It is clear from their structure that the above matrices ρai(Ul(i)) act on different spaces and hence they commute with each other. Let us now come to the study of the exact quantum symmetry algebras associated to the junction representation. Let us first briefly recall the Uq(sl2) algebra, defined by generators e, f , h that satisfy [13]: [h, e] = 2e, [h, f ] = −2f, [e, f ] = qh − q−h q − q−1 . The quantum algebra is as known equipped with a non-trivial co-product ∆ : Uq(sl2) 7→ Uq(sl2)⊗ Uq(sl2) ∆(qh) = qh ⊗ qh, ∆(x) = x⊗ q h 2 + q− h 2 ⊗ x, x ∈ {e, f}. The n co-product may be obtained by iteration as: ∆(n) = (I⊗∆(n−1))∆. Recall that the spin 1 2 representation of the algebra above is given by, πq : Uq(sl2) 7→ End(C2) πq(h) = σz, πq(e) = σ+, πq(f) = σ−, where σz, σ± are the 2× 2 Pauli matrices. Now define: π1(x) = πa1(x)⊗ I⊗ I, π2(x) = I⊗ πa2(x)⊗ I, π3(x) = I⊗ I⊗ πa3(x), x ∈ Uai(sl2). Concerning the junction representation, it is easy to check that:[ Θ(Ul), π⊗N i (∆(N)(x)) ] = 0, x ∈ Uai(sl2), i = 1, 2, 3. (5) It is obvious that each one of the representations acts on the spaces Vl(i) separately, that is[ ρai(Ul(i)), π ⊗N j (∆(N)(x)) ] = 0, i, j ∈ {1, 2, 3}, which immediately leads to (5). Hence, the junction representation enjoys a manifest G0 ≡ Ua1(sl2)⊗ Ua2(sl2)⊗ Ua3(sl2) symmetry, as is trivially expected by its construction. However, as in the case of the twin asymmetric representation [7, 6], our main objective is to extract non trivial quantum algebra realizations that commute with the junction representation. Consider the following representations fi : Uqi(sl2) 7→ End((C(2))⊗3), i = 0, 1, 2, 3 f0(h) = e11 ⊗ e11 ⊗ e11 − e22 ⊗ e22 ⊗ e22 = h (0) 1 − h (0) 2 , f0(e) = e12 ⊗ e12 ⊗ e12, f0(f) = e21 ⊗ e21 ⊗ e21, f0(qh 0 ) = I + (q0 − 1)h(0) 1 + ( q−1 0 − 1 ) h (0) 2 , (6a) f1(h) = e22 ⊗ e11 ⊗ e11 − e11 ⊗ e22 ⊗ e22 = h (1) 1 − h (1) 2 , f1(e) = e21 ⊗ e12 ⊗ e12, f1(f) = e12 ⊗ e21 ⊗ e21, f1(qh 1 ) = I + (q1 − 1)h(1) 1 + ( q−1 1 − 1 ) h (1) 2 , (6b) f2(h) = e11 ⊗ e22 ⊗ e11 − e22 ⊗ e11 ⊗ e22 = h (2) 1 − h (2) 2 , Junction Type Representations of the Temperley–Lieb Algebra 5 f2(e) = e12 ⊗ e21 ⊗ e12, f2(f) = e21 ⊗ e12 ⊗ e21, f2(qh 2 ) = I + (q2 − 1)h(2) 1 + ( q−1 2 − 1 ) h (2) 2 , (6c) f3(h) = e11 ⊗ e11 ⊗ e22 − e22 ⊗ e22 ⊗ e11 = h (3) 1 − h (3) 2 , f3(e) = e12 ⊗ e12 ⊗ e21, f3(f) = e21 ⊗ e21 ⊗ e12, f3(qh 3 ) = I + (q3 − 1)h(3) 1 + ( q−1 3 − 1 ) h (3) 2 . (6d) It is straightforward to check that each one of the fi’s defined above provide a representation of Uqi(sl2). After some quite cumbersome but straightforward computation, it is shown that[ Θ(Ul), f⊗N i (∆(N)(x)) ] = 0, x ∈ Uqi(sl2), i = 0, 1, 2, 3, (7) with qi to be determined by symmetry requirements. In Appendix B some explicit computations for i = 0 are presented. This in fact turns out to be an interesting combinatorics problem. Each symmetry is asso- ciated to a distinct qi, which is determined by symmetry requirements. In particular, we found the following values for qi, associated to the four representations fi: q0 = a1a2a3, q1 = a−1 1 a2a3, q2 = a1a −1 2 a3 q3 = a1a2a −1 3 (see also the Appendix A on the solution of the Hecke condition). To summarize, this is the announced non trivial symmetry of the junction representation. For the simplest case, n = 3, we have extracted the extra non-trivial symmetry of the junction representation, i.e. G ≡ Uq0(sl2)⊗ Uq1(sl2)⊗ Uq2(sl2)⊗ Uq3(sl2), essentially different from the manifest ⊗3 i=1 Uai(sl2) symmetry. It is important to note that this extra symmetry is structurally much richer than the manifest one, and is enlarged almost exponentially with respect to n, as will become transparent in the subsequent section. 2.2 Generic junction representation and exact symmetries We shall generalize here the investigation of the exact symmetries associated to the generic junction representation (n > 3). Inspired by the n = 3 case, and the respective form of the representations that have been obtained there, we propose a big family of representations below for generic values of n. Actually, after inspection, one observes that the whole analysis merely reduces to a combinatorial problem. Let us describe the generic family of representations of the quantum algebra. Consider first the representation f0 : Uq0(sl2) 7→ End((C(2))⊗n) as a starting point f0(h) = (e11)⊗n − (e22)⊗n ≡ h (0) 1 − h (0) 2 , f0(e) = (e12)⊗n, f0(f) = (e21)⊗n, f0(qh 0 ) = I + (q0 − 1)h(0) 1 + (q−1 0 − 1)h(0) 2 . Recall that in the generic situation, n constants ai are involved. The symmetry algebras exa- mined here are in fact associated to all possible combinations of ai’s and their inverses. More precisely, consider first the combination a1a2 · · · an, which is in fact equal to q0. By turning one of the ai to its inverse we obtain n such combinations. Then we may turn two, three ai’s etc. into their inverses, and stop turning after we reach the values n 2 or n−1 2 (for n even and odd respectively). We stop basically because we reproduce the same algebras as we shall see 6 A. Doikou and N. Karaiskos below. In fact, by repeating the procedure above we end up with a total number of combinations given by: M = n−1 2∑ m=0 n! m!(n−m)! if n odd, M = n 2 −1∑ m=0 n! m!(n−m)! + 1 2 n! n 2 !n 2 ! if n even. (8) We can now introduce the following generic family of representations fi1i2...im , m ∈ {0, . . . , L} where L = n 2 for n even and L = n−1 2 for n odd (for m = 0 we basically obtain f0). The corresponding deformation parameter is defined as qi1i2...im = (−1)n+1a1a2 · · · a−1 i1 ai1+1 · · · a−1 i2 · · · a−1 im · · · an, that is each one of the constants aij , ij ∈ {1, . . . , n} is turned to its inverse. It is thus clear that the total number of representations is given by the total number of combinations M given in (8). In fact, for every m we have n! m!(n−m)! number of representations, provided that for m = 0 there is only one representation, the f0. Note also that qi1...im = qP(i1...im), where P denotes all possible permutations among the elements ij . Notice that in the case of n even and m = n 2 we keep only half of the combinations. For instance take n = 4; we should only consider q34, q24, q23. It is clear that q12 = q−1 34 , f12(e) = f34(f), f12(f) = f34(e), and similarly for the rest (see also Appendix C for a more detailed description of the algebra elements). More generally, let us introduce the conjugate index: ī = n− i + 1. It is then clear that qi1...i n 2 = q−1 ī1...̄i n 2 . It is worth noting that some kind of duality is manifest throughout this process. In particular, the quantum algebra with parameter qi1...im is equivalent to the algebra with parameter q−1 i1...im , after interchanging e ↔ f and h ↔ −h. This is the reason why we stop the procedure of turning ai’s to their inverses after reaching the values n 2 or n−1 2 for n even and odd respectively, as described earlier in the text. Consequently, physical systems which may possess such underlying structures are expected to enjoy some kind of physical equivalence that reflect this duality. A generic representation is then defined along the lines described above as fi1i2...im : Uqi1i2...im (sl2) → End((C2)⊗n), fi1i2...im(h) = e11 ⊗ · · · e11 ⊗ · · · ⊗ e22︸︷︷︸ i1 ⊗e11 · · · ⊗ e22︸︷︷︸ i2 ⊗ · · · ⊗ e22︸︷︷︸ im ⊗ · · · ⊗ e11 − e22 ⊗ · · · e22 ⊗ · · · ⊗ e11︸︷︷︸ i1 ⊗e22 · · · ⊗ e11︸︷︷︸ i2 ⊗ · · · ⊗ e11︸︷︷︸ im ⊗ · · · ⊗ e22 ≡ h (i1...im) 1 − h (i1...im) 2 (9) fi1i2...im(e) = e12 ⊗ · · · e12 ⊗ · · · ⊗ e21︸︷︷︸ i1 ⊗e12 · · · ⊗ e21︸︷︷︸ i2 ⊗ · · · ⊗ e21︸︷︷︸ im ⊗ · · · ⊗ e12, Junction Type Representations of the Temperley–Lieb Algebra 7 fi1i2...im(f) = e21 ⊗ · · · e21 ⊗ · · · ⊗ e12︸︷︷︸ i1 ⊗e21 · · · ⊗ e12︸︷︷︸ i2 ⊗ · · · ⊗ e12︸︷︷︸ im ⊗ · · · ⊗ e21. All the above quantities emerge from f0 by exchanging: e11 ↔ e22 for h, e12 ↔ e21 for e, f, for every site ij ∈ {1, 2, . . . , n}. In a similar fashion the generalized representation for the qh element has the form fqi1...im ( qh i1...im ) = I + (qi1...im − 1)h(i1...im) 1 + ( q−1 i1...im − 1 ) h (i1...im) 2 . To prove that the above relations define a representation of the Uqi1...im (sl2) algebra is a straight- forward task. One just takes into account the basic property: eijekl = δjkeil. Having obtained this big family of Uqi1...im (sl2) representations, we conjecture that these are also non-trivial exact symmetries of the junction representation. More precisely, our main conjecture is that: [f⊗N i1...im (∆(N)(x)),Θ(Ul)] = 0, x ∈ Uqi1...im (sl2). (10) This is a plausible conjecture given the results for n = 3 and the particular structure of the mentioned representations. Note that we have also checked the validity of our conjecture for the cases n = 4, 5 with the use of the algebraic program Mathematica. Let us make some general qualitative comments that further enforce our claim. Consider first the multiplication1 Θ(U1)f⊗2 0 (∆(h)). Note that Θ contains a large number of different terms inside, which can be grouped into powers of a’s. (From now on and only for the following argument we let ai = ayi and count in powers of a.) Actually the number of terms of a specific power O(am) is given in terms of the binomial coefficient, but this is not so significant at the moment. On the other hand, it is instructive to note that the terms containing a’s are either e11 or e22, while those not containing a’s are e12 and e21. Consider now the multiplication of Θ with the first term of f⊗2 0 (∆(h)), that is with (e11)⊗n ⊗ I⊗n. Recall that Θ is naturally “broken” into two parts, each one living on (C2)⊗n. We see that in order for the product of Θ with the above term to be different from zero, the first part of Θ should only contain the basis vectors e11 and e21, in every possible combination. Otherwise it just gives zero. Consequently, the problem is reduced into a combinatorics problem. However, for the case of f⊗2 0 (∆(h)) the situation is easy to handle2, and we see that the product Θ(U1)f⊗2 0 (∆(h)) vanishes identically. The reason is the following. We observe that from the multiplication of Θ with the first term of f⊗2 0 (∆(h)) only some specific terms survive. However, these are exactly the same terms that survive the multiplication of Θ with the last term of the co-product, namely with I⊗n ⊗ (e22)⊗n. And since those two terms appear with a different sign, they cancel out. The same happens with the other two terms. The same argument is also valid for the second term of the commutator, 1We focus here on the f0 representation but our arguments may be applied for the generic family of represen- tations. 2For the co-products of the other two elements the situation is more difficult to handle, since the products with Θ do not vanish identically. Hence the various conditions relating ai’s must be taken into account, which makes the presentation of a general argument a difficult task. 8 A. Doikou and N. Karaiskos that is for the multiplication f⊗2 0 (∆(h))Θ, which is identically zero. Hence, we have shown – using a somehow heuristic argument – that[ Θ(U1), f⊗2 0 (∆(h)) ] = 0. Note that this result is valid for all the representations that belong to the generic family that we have considered so far, due to the highly symmetric nature of f⊗2 i1...im (∆(h)). It should be stressed that although the above argument is not a rigorous proof, we expect that it holds for the whole family of representations. Similar arguments, albeit more complicated in their technicalities, can be presented for the other two elements of the quantum algebra. Finally, let us briefly discuss for completeness the existence of trivial symmetries associated to the junction representation. Define: πi(x) = I⊗ · · · ⊗ · · ·πai(x)︸︷︷︸ 1(i) ⊗ · · · ⊗ I, x ∈ Uai(sl2). (11) It is straightforward then to check that the junction representation commutes with the actions defined above. Each one of the representations ρai acts non trivially on the spaces C2 l(i) ⊗C2 (l+1)(i) separately, that is[ ρai(Ul(i)), π ⊗N j (∆(N)(x)) ] = 0, i, j ∈ {1, . . . , n}, which immediately leads to[ Θ(Ul), π⊗N i (∆(N)(x)) ] = 0, x ∈ Uai(sl2), (12) exposing the manifest symmetry of a generic junction representation G0 ≡ Ua1(sl2)⊗ Ua2(sl2)⊗ · · · ⊗ Uan(sl2). (13) To summarize, we have been able to explicitly prove the existence of non-trivial exact sym- metries (6), (7) associated to the junction representation for n = 3 (see Appendix B). We have conjectured the generic form of non-trivial symmetries of the junction representation (9) ∀n, and we have checked the validity of our conjecture (10) with Mathematica for n = 4, 5. The existence of trivial symmetries (11) is an immediate consequence of the structure of the junction representation, therefore the proof of (12), (13) is straightforward ∀n. 3 Boundary Temperley–Lieb algebra and representations We shall discuss in this section extensions of the junction representation in the case of the boundary Temperley–Lieb algebra. The boundary Temperley–Lieb BN (q, Q) algebra is defined by generators Ui obeying (1) and U0 satisfying U2 0 = δ0U0, U1U0U1 = κU1, [U0, Ui] = 0, i > 1, where δ0 = −(Q + Q−1), κ = qQ−1 + q−1Q, although there is a reparametrization freedom that one can appropriately use (see e.g. [7]). The well known XXZ representation of the boundary Temperley–Lieb algebra is given next. Let U be the matrix defined in (2) and U0 = −Q−1e11 −Qe22 + e12 + e21. (14) Junction Type Representations of the Temperley–Lieb Algebra 9 Then the following is a representation of the boundary TL algebra, ρq,Q : BN (q, Q) 7→ End ( (C2)⊗N ) , ρq,Q(Ui) = I⊗ · · · ⊗ I⊗ U︸︷︷︸ i, i+1 ⊗I⊗ · · · ⊗ I, ρq,Q(U0) = U0 ⊗ I⊗ · · · ⊗ I. (15) The obvious junction representation for the “boundary” element for generic n is then the fol- lowing Θ(U0) = n∏ i=1 ρai,Qi(U0), where each one of the XXZ representations act on the spaces C2 1(1) , C2 1(2) , . . ., C2 1(n) ; i.e. ρai,Qi(U0) = I⊗ · · · ⊗ U0︸︷︷︸ 1(i) ⊗ · · · ⊗ I. It is straightforward to check, due to the fact that spaces C2 l(i) commute among each other that Θ(Ui) and Θ(U0) do satisfy the algebraic relations of the boundary TL algebra, thus they provide a representation of the algebra. However, our objective here is to search for non trivial boundary elements that mix essentially the spaces exactly as the non trivial symmetries mix the various spaces C2 1(i) . Consider the following matrices Mi1...im = −Q−1h (i1...im) 1 −Qh (i1...im) 2 + fi1...im(e) + fi1...im(f), and recall that h (i1...im) 1,2 and fi1...im(e, f) are defined in Section 2.2. The indices take values ij ∈ {1, . . . , n} as also described in Section 2.2. Let us focus again on the n = 3 case, where analytic proofs of our claims are available (for higher n analytic proofs are highly complicated, and only checks with the help of algebraic packages can be performed). First, introduce the following matrices corresponding to the four fi representations of the n = 3 case described in Section 2.1, M0 = −Q−1e11 ⊗ e11 ⊗ e11 −Qe22 ⊗ e22 ⊗ e22 + e12 ⊗ e12 ⊗ e12 + e21 ⊗ e21 ⊗ e21, M1 = −Q−1e22 ⊗ e11 ⊗ e11 −Qe11 ⊗ e22 ⊗ e22 + e21 ⊗ e12 ⊗ e12 + e12 ⊗ e21 ⊗ e21, M2 = −Q−1e11 ⊗ e22 ⊗ e11 −Qe22 ⊗ e11 ⊗ e22 + e12 ⊗ e21 ⊗ e12 + e21 ⊗ e12 ⊗ e21, M3 = −Q−1e11 ⊗ e11 ⊗ e22 −Qe22 ⊗ e22 ⊗ e11 + e12 ⊗ e12 ⊗ e21 + e21 ⊗ e21 ⊗ e12. The matrices Mi act on C2 1(1) ⊗ C2 1(2) ⊗ C2 1(3) . Also set Mi = Mi ⊗ I⊗ · · · ⊗ I, where now I acts on (C2)⊗3. It can be shown then that Θ(U1)MiΘ(U1) = κΘ(U1), i = 0, 1, 2, 3, and this suffices to prove that this is a representation of the boundary Temperley–Lieb algebra. Indeed we define the representation Θi : BN (q, Q) 7→ End(C2⊗C2⊗C2)⊗N , i ∈ {0, 1, 2, 3} such that: Θi(Ul) = Θ(Ul), l ∈ {1, 2, . . . , N − 1}, Θi(U0) = Mi. 10 A. Doikou and N. Karaiskos We can proceed then and state the following generic conjecture ∀n > 3. We conjecture that Θi1...im such that Θi1...im(Ul) = Θ(Ul), l ∈ {1, . . . , N − 1}, Θi1...im(U0) = Mi1...im , Mi1...im = Mi1...im ⊗ I⊗ · · · ⊗ I is a representation of the boundary Temperley–Lieb algebra. To prove this it is sufficient to show that Θ(U1)Mi1...imΘ(U1) = κΘ(U1). As before, we have checked the validity of the latter conjecture by performing an analytical computation for n = 3 and by using Mathematica for the cases n = 4, 5. Therefore, we have strong indications that it is valid for generic n. The analytic proof for n = 3 is omitted here for brevity, but it is straightforward, although technically demanding, and is based on the explicit form of Θ(U1), Mi1...im and the property: eijekl = δjkeil As mentioned above, the structure of these boundary elements is inspired by the form of the non trivial symmetries studied in the previous section. It will be transparent in the subsequent section that they break the respective symmetries then, along the lines described in [7, 8, 6]. 3.1 Residual symmetries In this section we shall examine how each one of the non-trivial boundary elements Mi1...im breaks the original symmetry of the Temperley–Lieb algebra. This will be particularly useful when studying the corresponding quantum spin chain symmetry. Apart from the manifest symmetry of the algebra, we have exposed in the previous section a Uq0(sl2)⊗Uq1(sl2)⊗Uq2(sl2)⊗Uq3(sl2) symmetry for the junction representation. It is straight- forward to check that each one of the non-trivial boundary elements breaks only part of this symmetry, while preserving the rest of it. In particular, it is shown that [fi(x),Mj ] = 0, x ∈ Uqi(sl2), i 6= j. The above condition is not accidental at all. In fact, it is somehow expected, given that the structure of those boundary elements is strictly related to the respective non-trivial symmetries of the algebra. However, the case where i = j requires particular attention. Let us first consider the following combination of generators of the quantum algebra Uqi1...im (sl2) (see also e.g. [6, 19, 5]) Qi1...im = q − 1 2 i1...im q h 2 i1...im e + q 1 2 i1...im q −h 2 i1...im f + xi1...imqh i1...im − xi1...imI, where xi1...im are constants to be determined later on. Focus again on the simplest non-trivial situation n = 3. Using then the four fi representations at hand one may compute the respective charges, which have the following form f0(Q0) = (−x0 + q0x0)e11 ⊗ e11 ⊗ e11 + e12 ⊗ e12 ⊗ e12 + e21 ⊗ e21 ⊗ e21 + (−x0 + q−1 0 x0)e22 ⊗ e22 ⊗ e22, f1(Q1) = (−x1 + q1x1)e22 ⊗ e11 ⊗ e11 + e21 ⊗ e12 ⊗ e12 + e12 ⊗ e21 ⊗ e21 + (−x1 + q−1 1 x1)e11 ⊗ e22 ⊗ e22, f2(Q2) = (−x2 + q2x2)e11 ⊗ e22 ⊗ e11 + e12 ⊗ e21 ⊗ e12 + e21 ⊗ e12 ⊗ e21 + (−x2 + q−1 2 x2)e22 ⊗ e11 ⊗ e22, Junction Type Representations of the Temperley–Lieb Algebra 11 f3(Q3) = (−x3 + q3x3)e11 ⊗ e11 ⊗ e22 + e12 ⊗ e12 ⊗ e21 + e21 ⊗ e21 ⊗ e12 + (−x3 + q−1 3 x3)e22 ⊗ e22 ⊗ e11. This charge commutes with all the irrespective boundary elements, that is [fi(Qi),Mj ] = 0 for i 6= j, but is also found to commute with the relevant boundary element [fi(Qi),Mi] = 0, provided that the constants xi satisfy xi = Q−Q−1 qi − q−1 i . To conclude, the symmetry we extract here for each representation Θi of the boundary Temper- ley–Lieb algebra is:[ Θi(Ul), f⊗N i (∆(N)(Qi))⊗i6=j f⊗N j (∆(N)(x)) ] = 0, x ∈ Uqj (sl2). We may generalize the statement – as a conjecture – on the boundary symmetry ∀n, i.e., we conjecture that [Θi1...im(Ul), f⊗N i1...im (∆(N)(Qi1...im))⊗{i1,...,im}6={j1,...,jm} f⊗N j1...jm (∆(N)(x))] = 0, x ∈ Uqj1...jm (sl2). (16) It is clear that the trivial representation defined in (14), (15) enjoys the symmetry (see also [6]):[ Θi(Ul), π⊗N i (∆(N)(Qi))⊗i6=j π⊗N j (∆(N)(x)) ] = 0, x ∈ Uaj (sl2), i, j ∈ {1, . . . , n}, where we define Qi = a − 1 2 i a h 2 i e + a 1 2 i a −h 2 i f + xia h i − xiI, i ∈ {1, . . . , n}, and xi = Q−Q−1 ai − a−1 i . For a more detailed discussion on boundary symmetries we refer the interested reader to [7, 6]. 4 The quantum spin chain We shall now briefly discuss the corresponding quantum spin chain, enjoying the symmetries described above. Recall the Ř matrix is a solution of the braid Yang–Baxter equation Ř12(λ1 − λ2)Ř23(λ1)Ř12(λ2) = Ř23(λ2)Ř12(λ1)Ř23(λ1 − λ2). The Ř matrix associated with representations of the Temperley–Lieb algebra may be expressed as [12] Řii+1(λ) = sinh(λ + iµ) + sinhλρ(Ui), for any representation ρ of the TL algebra TN (q), and where q = eiµ. 12 A. Doikou and N. Karaiskos Define the R matrix as: R(λ) = PŘ(λ). The R matrix associated with the junction repre- sentation satisfies unitarity and crossing symmetry i.e. Rij(λ)Rji(−λ) ∝ I, Rij(λ) ∝ ViR tj ij(−λ− i)Vi, where we define in general V = ⊗n i=1 ( a − 1 2 i a 1 2 i ) , and for n = 3 in particular we have: V = ( a − 1 2 1 a 1 2 1 ) ⊗ ( a − 1 2 2 a 1 2 2 ) ⊗ ( a − 1 2 3 a 1 2 3 ) =  q − 1 2 0 q − 1 2 3 q − 1 2 2 q 1 2 1 q − 1 2 1 q 1 2 2 q 1 2 3 q 1 2 0  . Recall also the identifications of qi’s, entailed by the exact symmetries of the representation. Then one may construct the transfer matrix of an open spin chain with generic integrable boundary conditions [23]: t(λ) = tr0{M0K + 0 T(λ)}, where T = T0(λ)K0(λ)T−1 0 (λ), T (λ) = R0N (λ) · · ·R01(λ). The K matrix satisfies the reflection equation R12(λ1 − λ2)K1(λ1)R21(λ1 + λ2)K2(λ2) = K2(λ2)R12(λ1 + λ2)K1(λ1)R21(λ1 − λ2). We shall focus here on the case of trivial left boundary, that is K+ ∝ I. Also M is defined as M = VtV, and for n = 3 in particular it becomes: M =  q0 q3 q2 q−1 1 q1 q−1 2 q−1 3 q−1 0  . Junction Type Representations of the Temperley–Lieb Algebra 13 It is known that solutions of the reflection equation may be expressed in terms of representations of the boundary Temperley–Lieb algebra as (see also e.g. [16, 7]): K(λ) = x(λ) + y(λ)ρ(U0), x(λ) = −δ0 cosh(2λ + iµ)− κ cosh(2λ)− cosh(2iζ), y(λ) = 2 sinh(2λ) sinh(iµ). Let us first consider the case of a trivial right boundary i.e. K ∝ I. Then as was proven in Section 2.1 that the junction representation enjoys the ⊗3 i=0Uqi(sl2) symmetry and consequently[ Ř, f⊗2 i (∆⊗2(x)) ] = 0 ⇒ f⊗2 i (∆ ′⊗2(x))R(λ) = R(λ)f⊗2 i (∆⊗2(x)), x ∈ Uqi(sl2), where ∆′ = P∆P and where R is the junction R matrix. From the relations above it is straightforward to show that: f⊗(N+1) i (∆ ′⊗(N+1)(x))T (λ) = T (λ)∆⊗(N+1)(f⊗(N+1) i (x)), which leads, for trivial boundary conditions to f⊗(N+1) i (∆ ′⊗(N+1)(x))T(λ) = T(λ)f⊗(N+1) i (∆ ′⊗(N+1)(x)) ⇒ [t(λ), f⊗(N) i (∆⊗(N)(x))] = 0. Hence, the transfer matrix with trivial boundary conditions enjoys the ⊗3 i=0Uqi(sl2) symmetry (similarly it may be shown that the transfer matrix enjoys the obvious symmetry ⊗3 i=1Uai(sl2), see also [8, 6] for further details). The statement may be generalized for any fi1...im representation ∀n provided that relations (10) are valid:[ t(λ), f⊗(N) i1...im (∆⊗(N)(x)) ] = 0, x ∈ Uqi1...im (sl2). We shall focus here, since it is more interesting, on the breaking of the non trivial symmetries expressed via the representations fi. The existence of a non-trivial “boundary” may couple the various Vl(i) spaces. In this case various choices of representations of the boundary Temperley– Lieb algebra lead to consistent breaking of the original symmetry in different ways. Particularly, focus first on the n = 3 case and define K(i) = x(λ) + y(λ)Mi and T(i) = T (λ)K(i)(λ)T−1(−λ). It is clear bearing also in mind the results of Section 3 that[ K(i)(λ), fi(Qi)⊗j 6=i fj(x) ] = 0 ⇒ T(i)(λ)f⊗(N+1) i (∆ ′(N+1)(Qi))⊗j 6=i f⊗(N+1) j (∆ ′(N+1)(x)) = f⊗(N+1) i (∆ ′(N+1)(Qi))⊗j 6=i f⊗(N+1) j (∆ ′(N+1)(x))T(i)(λ), and it is then shown that[ t(i)(λ), f⊗N i (∆(N)(Qi))⊗j 6=i f⊗N j (∆(N)(x)) ] = 0, x ∈ Uqj (sl2). With this statement we basically conclude our study on the symmetries of the junction spin chain with trivial and non-trivial integrable boundary conditions. Again the boundary symmetry statement can be generalized for any representation fi1...im , ∀n provided that relations (10), (16) hold. Indeed, set K(i1...im) = x(λ) + y(λ)Mi1...im , t(i1...im)(λ) = tr{MT (λ)K(i1...im)(λ)T−1(−λ)}. 14 A. Doikou and N. Karaiskos Then it is straightforward to show, provided that (10), (16) are valid:[ t(i1...im)(λ), f⊗N i1...im (∆(N)(Qi1...im))⊗{j1,...,jm}6={i1,...im} f⊗N j1...jm (∆(N)(x)) ] = 0, x ∈ Uqj1...jm (sl2). Since the junction representation is a representation of the (boundary) TL algebra as is the XXZ one expects, according to the general discussion in [7], that the spectra of the models will by identical, up to multiplicities, (see universal expressions for Hamiltonians in [7] and generic expressions of open transfer matrices in terms of elements of the (boundary) Temperley–Lieb algebra in [6]). Note that the results of the present section are explicitly proven provided that the results, and conjectures of Sections 2.1, 2.2, 3 and 3.1 are valid. 5 Discussion We have been able to identify a novel family of representations of the Temperley–Lieb algebra, the so called junction representations, which involve n copies of the familiar XXZ representation. Interestingly enough, it turns out that the number of exact symmetries associated to the junc- tion representation is drastically increased with the number n. The existence of the extended junction representation in the case of the boundary Temperley–Lieb algebra is examined, and particular non-trivial “boundary” elements are identified. The results on the boundary case are non-trivial and analytic proof is provided in the case where n = 3, whereas in the general case a conjecture on the form of the representation of the boundary element is stated, and is checked with Mathematica for n = 4, 5. The associated residual symmetries in this case are also extracted. More precisely, regarding the study of the associated symmetries: we explicitly show in Section 2.1 that in addition to the expected symmetries particular non-trivial algebra realizations commute with the junction representation, whereas in Section 3 we state a generic conjecture on the existence of a big number of non-trivial symmetry algebras, which we support qualitatively. We check however the validity of our conjecture with Mathematica up to n = 5. Similar results obtained in Section 3.1 concerning the associated boundary symmetries. What is highly non-trivial in the present investigation in the existence of a large number of non-trivial exact symmetry algebras. It is important to note that for the moment there is no-rigorous proof that these consist the full symmetry algebra associated to the junction representation. Nevertheless, it would be very interesting if one could find an extra symmetry that does not fall to the categories discussed in the present study. Moreover, the obvious number of associated symmetries does not seem to coincide with the number of the obtained non-trivial symmetries, and this is also an intriguing issue, which is still open even in the simplest case n = 2, the “asymmetric twin” representation [7]. We hope to address these significant issues in future investigations. Using this particular representation we constructed the corresponding novel physical quan- tum system, that is the “junction” quantum spin chain, and examined the associated exact symmetries. Within this spirit an interesting problem to pursue is the study of the aforemen- tioned representations, and symmetries in the context of other statistical physics, such as face or loop models. An even more intriguing question is the possible existence of some kind of con- tinuum limit of the junction spin chain, which also enjoys the big number of exact symmetries identified here. More precisely, universal local Hamiltonians in terms of the elements of the (boundary) Temperley–Lieb algebras are available (see e.g. [7, 8]), thus one may try to obtain the so called long wave length continuum limit (see e.g. [9]) of such expressions, that is obtain classical continuum Hamiltonians. We shall fully address these and related issues in forthcoming publications. Junction Type Representations of the Temperley–Lieb Algebra 15 A Parametrization and solution of the Hecke condition In this appendix we present how one obtains solutions of the quadratic constraint of the Temperley–Lieb algebra (Hecke condition) U2 = − ( q + q−1 ) U. We focus first for simplicity on the case n = 3. A detailed study for the generic solution will be presented elsewhere. By taking into account the form of the junction representation, the Hecke condition is written as a1a2a3 + a−1 1 a−1 2 a−1 3 + a1a2a −1 3 + a1a −1 2 a3 + a−1 1 a2a3 + a−1 1 a−1 2 a3 + a−1 1 a2a −1 3 + a1a −1 2 a−1 3 = q + q−1. It is now convenient to parametrize ai as follows: ai = ziq 1 3 , i ∈ {1, 2, 3} In addition, if we set a1a2a3 = q ⇒ a−1 1 a−1 2 a−1 3 = q−1, then the rest terms sum up to give zero. Hence we are left with three equations to solve in order to identify the three unknown quantities zi, which read z1z2z3 = 1, z1z2z −1 3 + z1z −1 2 z3 + z−1 1 z2z3 = 0 z−1 1 z−1 2 z3 + z−1 1 z2z −1 3 + z1z −1 2 z−1 3 = 0. (17) Solving the equations above one can explicitly fix the values of zi, for the particular parametriza- tion of ai. Of course one may choose a different parametrization, but in any case these should be equivalent. A simple solution of the above system of equations is the following: z1 = e iπ 3 , z2 = e 2iπ 3 , z3 = eiπ. It is straightforward to check that the above solution satisfies the system (17). Let us now generalize our comments for every n > 3. We start from the generic polynomial condition (−1)n n∏ j=1 ( aj + a−1 j ) = − ( q + q−1 ) , and expanding the expressions above we take a1a2 · · · an + ∑ i a1 · · · a−1 i · · · an + ∑ i1,i2 a1 · · · a−1 i1 · · · a−1 i2 · · · an + · · · = (−1)n−1 ( q + q−1 ) . We parameterize, as in n = 3 case, by setting q = (−1)n−1a1a2 · · · an, aj = zjq 1 n . (18) Our task now is to identify the constants zi. By also requiring, in addition to (18), the factor in front of every power q m n (−n < m < n) to disappear we end up with a set of n equations. More specifically, we end up with a system of n unknown quantities, the zj ’s and n equations, so that in principle the system can be solved and zj ’s can be determined. 16 A. Doikou and N. Karaiskos B Proof for n = 3 As it has been already stated, we have checked the validity of our conjecture for the cases n = 4, 5 by using Mathematica. However, we were able to perform analytic calculations for the n = 3 case. In this appendix, we describe part of our analytic computation. It is convenient for what follows to write down the explicit expression of the each one of the XXZ representations for n = 3: i.e. ρa1(Ul(1)) = · · · I⊗ (∑ a 6=b eab ⊗ I⊗ I⊗ eba ⊗ I⊗ I − ∑ a 6=b a −sgn(a−b) 1 eaa ⊗ I⊗ I⊗ ebb ⊗ I⊗ I ) ⊗ I · · · , ρa2(Ul(2)) = · · · I⊗ (∑ a 6=b I⊗ eab ⊗ I⊗ I⊗ eba ⊗ I − ∑ a 6=b a −sgn(a−b) 2 I⊗ eaa ⊗ I⊗ I⊗ ebb ⊗ I ) ⊗ I · · · , ρa3(Ul(3)) = · · · I⊗ (∑ a 6=b I⊗ I⊗ eab ⊗ I⊗ I⊗ eba − ∑ a 6=b a −sgn(a−b) 3 I⊗ I⊗ eaa ⊗ I⊗ I⊗ ebb ) ⊗ I · · · . Our purpose is to show that[ Θ(Ul), f⊗N i (∆(x)) ] = 0, x ∈ Uqi(sl2), for i = 0, . . . , 3. Let us focus for simplicity, and for saving writing on N = 2, since it is then easy to generalize for any N (see also [7]). We shall show in detail the computation for the f0 representation, the rest follow in the same spirit. Consider first the element e of the Uq0(sl2) quantum algebra. The left-hand side of [Θ(U1), f⊗2 0 (∆(e))] is written explicitly as Θ(U1)f⊗2 0 (∆(e)) = (∑ a 6=b eab ⊗ I⊗ I⊗ eba ⊗ I⊗ I− ∑ a 6=b a −sgn(a−b) 1 eaa ⊗ I⊗ I⊗ ebb ⊗ I⊗ I ) × (∑ m6=n I⊗ emn ⊗ I⊗ I⊗ enm ⊗ I− ∑ m6=n a −sgn(m−n) 2 I⊗ emm ⊗ I⊗ I⊗ enn ⊗ I ) × (∑ k 6=l I⊗ I⊗ ekl ⊗ I⊗ I⊗ elk − ∑ k 6=l a −sgn(k−l) 3 I⊗ I⊗ ekk ⊗ I⊗ I⊗ ell ) × (( q − e11 2 0 ⊗ e11 ⊗ e11 + q e22 2 0 ⊗ e22 ⊗ e22 + I⊗ e11 ⊗22 +I⊗ e22 ⊗ e11 ) ⊗ e12 ⊗ e12 ⊗ e12 +⊗e12 ⊗ e12 ⊗ e12 ⊗ ( q e11 2 0 ⊗ e11 ⊗ e11 + q − e22 2 0 ⊗ e22 ⊗ e22 + I⊗ e11 ⊗22 +I⊗ e22 ⊗ e11 )) . Using the property eijekl = δjkeil and after some quite cumbersome algebra we end up with the following expression for the first Θ(U1)f⊗2 0 (∆(e)) = q 1 2 0 e12 ⊗ e12 ⊗ e12 ⊗ e22 ⊗ e22 ⊗ e22 Junction Type Representations of the Temperley–Lieb Algebra 17 + q − 1 2 0 e22 ⊗ e22 ⊗ e22 ⊗ e12 ⊗ e12 ⊗ e12 − a−1 3 q 1 2 0 e12 ⊗ e12 ⊗ e22 ⊗ e22 ⊗ e22 ⊗ e12 − a3q − 1 2 0 e22 ⊗ e2 ⊗ e12 ⊗ e12 ⊗ e12 ⊗ e22 − a−1 2 q 1 2 0 e12 ⊗ e22 ⊗ e12 ⊗ e22 ⊗ e12 ⊗ e22 − a2q − 1 2 0 e22 ⊗ e12 ⊗ e22 ⊗ e12 ⊗ e22 ⊗ e12 + a−1 2 a−1 3 q 1 2 0 e12 ⊗ e22 ⊗ e22 ⊗ e22 ⊗ e12 ⊗ e12 + a2a3q − 1 2 0 e22 ⊗ e12 ⊗ e12 ⊗ e12 ⊗ e22 ⊗ e22 − a−1 1 q 1 2 0 e22 ⊗ e12 ⊗ e12 ⊗ e12 ⊗ e22 ⊗ e22 − a1q − 1 2 0 e12 ⊗ e22 ⊗ e22 ⊗ e22 ⊗ e12 ⊗ e12 + a−1 1 a−1 3 q 1 2 0 e22 ⊗ e12 ⊗ e22 ⊗ e12 ⊗ e22 ⊗ e12 + a1a2q − 1 2 0 e12 ⊗ e22 ⊗ e12 ⊗ e22 ⊗ e12 ⊗ e12 + a−1 1 a−1 2 q 1 2 0 e22 ⊗ e22 ⊗ e12 ⊗ e12 ⊗ e12 ⊗ e22 + a1a2q − 1 2 0 e12 ⊗ e12 ⊗ e22 ⊗ e22 ⊗ e22 ⊗ e12 − a−1 1 a−1 2 a−1 3 q 1 2 0 e22 ⊗ e22 ⊗ e22 ⊗ e12 ⊗ e12 ⊗ e12 − a1a2a3q − 1 2 0 e12 ⊗ e12 ⊗ e12 ⊗ e22 ⊗ e22 ⊗ e22. (19) and the second term of the commutator f⊗2 0 (∆(e))Θ(U1) = q − 1 2 0 e12 ⊗ e12 ⊗ e12 ⊗ e11 ⊗ e11 ⊗ e11 + q 1 2 0 e11 ⊗ e11 ⊗ e11 ⊗ e12 ⊗ e12 ⊗ e12 − a3q − 1 2 0 e12 ⊗ e12 ⊗ e11 ⊗ e11 ⊗ e11 ⊗ e12 − a−1 3 q 1 2 0 e11 ⊗ e11 ⊗ e12 ⊗ e12 ⊗ e12 ⊗ e11 − a2q − 1 2 0 e12 ⊗ e11 ⊗ e12 ⊗ e11 ⊗ e12 ⊗ e11 − a−1 2 q 1 2 0 e11 ⊗ e12 ⊗ e11 ⊗ e12 ⊗ e11 ⊗ e12 + a2a3q − 1 2 0 e12 ⊗ e11 ⊗ e11 ⊗ e11 ⊗ e12 ⊗ e12 + a−1 2 a−1 3 q 1 2 0 e11 ⊗ e12 ⊗ e12 ⊗ e12 ⊗ e11 ⊗ e11 − a1q − 1 2 0 e11 ⊗ e12 ⊗ e12 ⊗ e12 ⊗ e11 ⊗ e11 − a−1 1 q 1 2 0 e12 ⊗ e11 ⊗ e11 ⊗ e11 ⊗ e12 ⊗ e12 + a1a3q − 1 2 0 e11 ⊗ e12 ⊗ e11 ⊗ e12 ⊗ e11 ⊗ e12 + a−1 1 a1 3q 1 2 0 e12 ⊗ e11 ⊗ e12 ⊗ e11 ⊗ e12 ⊗ e11+a1a2q − 1 2 0 e11 ⊗ e11 ⊗ e12 ⊗ e12 ⊗ e12 ⊗ e11 + a−1 1 a−1 2 q 1 2 0 e12 ⊗ e12 ⊗ e11 ⊗ e11 ⊗ e11 ⊗ e12 − a1a2a2q − 1 2 0 e11 ⊗ e11 ⊗ e11 ⊗ e12 ⊗ e12 ⊗ e12 − a−1 1 a−1 2 a−1 3 q 1 2 0 e12 ⊗ e12 ⊗ e12 ⊗ e11 ⊗ e11 ⊗ e11. (20) Now notice that every two terms in each side of the commutator are canceled provided that q0 = a1a2a3. This result has also been found to hold for the n = 4, 5 cases; each side of the commutator vanishes identically, for the appropriate form of qi, qi1i2 . The commutator of the junction representation with f⊗2 0 (∆(f)) follows in the same spirit and the expressions obtained are very similar to those presented in (19) and (20). However, the commutator of the representation with f⊗2 0 (∆(h)) is even easier to cope with, since it gives identically zero, without requiring any particular form for the qi. This has also been outlined in the general comments of Section 2.2, where the relation [Θ(U1), f⊗N i1...im (∆(N)(h))] = 0 has been proved there by using some simple heuristic arguments. In the same spirit the proof can be extended for i = 1, 2, 3 (n = 3). C Explicit expressions for n = 4 We present in this Appendix explicit expressions of the representations for the n = 4 case to further illustrate the structure of these representations. We also write down the explicit value 18 A. Doikou and N. Karaiskos of qi, qij so that the respective representation is a quantum symmetry of our algebra f0(e) = e12 ⊗ e12 ⊗ e12 ⊗ e12, f0(f) = e21 ⊗ e21 ⊗ e21 ⊗ e21, f0(h) = e11 ⊗ e11 ⊗ e11 ⊗ e11 − e22 ⊗ e22 ⊗ e22 ⊗ e22, q0 = −a1a2a3a4, f1(e) = e21 ⊗ e12 ⊗ e12 ⊗ e12, f1(f) = e12 ⊗ e21 ⊗ e21 ⊗ e21, f1(h) = e22 ⊗ e11 ⊗ e11 ⊗ e11 − e11 ⊗ e22 ⊗ e22 ⊗ e22, q1 = −a−1 1 a2a3a4, f2(e) = e12 ⊗ e21 ⊗ e12 ⊗ e12, f2(f) = e21 ⊗ e12 ⊗ e21 ⊗ e21, f2(h) = e11 ⊗ e22 ⊗ e11 ⊗ e11 − e22 ⊗ e11 ⊗ e22 ⊗ e22, q2 = −a1a −1 2 a3a4, f3(e) = e12 ⊗ e12 ⊗ e21 ⊗ e12, f3(f) = e21 ⊗ e21 ⊗ e12 ⊗ e21, f3(h) = e11 ⊗ e11 ⊗ e22 ⊗ e11 − e22 ⊗ e22 ⊗ e11 ⊗ e22, q3 = −a1a2a −1 3 a4, f4(e) = e12 ⊗ e12 ⊗ e12 ⊗ e21, f4(f) = e21 ⊗ e21 ⊗ e21 ⊗ e12, f4(h) = e11 ⊗ e11 ⊗ e11 ⊗ e22 − e22 ⊗ e22 ⊗ e22 ⊗ e11, q4 = −a1a2a3a −1 4 , f34(e) = e12 ⊗ e12 ⊗ e21 ⊗ e21, f34(f) = e21 ⊗ e21 ⊗ e12 ⊗ e12, f34(h) = e11 ⊗ e11 ⊗ e22 ⊗ e22 − e22 ⊗ e22 ⊗ e11 ⊗ e11, q34 = −a1a2a −1 3 a−1 4 , f24(e) = e12 ⊗ e21 ⊗ e12 ⊗ e21, f24(f) = e21 ⊗ e12 ⊗ e21 ⊗ e12, f24(h) = e11 ⊗ e22 ⊗ e11 ⊗ e22 − e22 ⊗ e11 ⊗ e22 ⊗ e11 q24 = −a1a −1 2 a3a −1 4 , f23(e) = e12 ⊗ e21 ⊗ e21 ⊗ e12, f23(f) = e21 ⊗ e12 ⊗ e12 ⊗ e21, f23(h) = e11 ⊗ e22 ⊗ e22 ⊗ e11 − e22 ⊗ e11 ⊗ e11 ⊗ e22, q23 = −a1a −1 2 a−1 3 a4. 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Junction Type Representations of the Temperley-Lieb Algebra and Associated Symmetries

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