Paths on graphs and associated quantum groupoids

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(1)REVISTA DE LA UNIÓN MATEMÁTICA ARGENTINA Volumen 51, Número 2, 2010, Páginas 147–170. PATHS ON GRAPHS AND ASSOCIATED QUANTUM GROUPOIDS R. TRINCHERO Abstract. Given any simple biorientable graph it is shown that there exists a weak *-Hopf algebra constructed on the vector space of graded endomorphisms of essential paths on the graph. This construction is based on a direct sum decomposition of the space of paths into orthogonal subspaces one of which is the space of essential paths. Two simple examples are worked out with certain detail, the ADE graph A3 and the affine graph A[2] . For the first example the weak *-Hopf algebra coincides with the so called double triangle algebra. No use is made of Ocneanu’s cell calculus.. 1. Introduction One of the most interesting developments in mathematical physics of the last decades has been the classification of SU(2)-type rational conformal field theories by ADE graphs1[1]. In relation to the present work a possible way to look at this classification is the following2. The tensor category of representations of a weak *Hopf algebra[5] constructed out of the corresponding ADE graph G is summarized by another graph Oc(G), called the Ocneanu graph of quantum symmetries[6]. Knowledge of this last graph encodes information on the conformal field theory when considered in various environments, the corresponding generalized partition functions can be obtained from this graph[1, 7, 8]. In addition the weak *-Hopf algebras mentioned above can be given a physical interpretation as the algebras of quantum mechanical symmetries of certain quantum statistical models, known as face models[9]. For the case of ADE graphs the weak *-Hopf algebra mentioned above is known as the the double triangle algebra(DTA)[6, 10, 11]. The construction of this algebra out of the corresponding ADE graph starts from something called quantum 6-j symbols[12] that can be computed employing Ocneanu’s cell calculus. These objects describe the representation theory of the DTA[13, 14, 15]. No direct derivation of. CONICET support is gratefully acknowledged. 1 Analog classifications exist for SU(3)-type[2] and SU(4)-type[3] rational conformal field theories, however the construction of the corresponding weak Hopf algebras out of the analog of the ADE graphs is not known. 2 Another way closer to the historical path is given in [4].. 147.

(2) 148. R. TRINCHERO. this weak Hopf algebra out of paths on the corresponding graph is available in the literature. One of the aims of this work is to fill this gap3. The key ingredient in this derivation is a direct sum decomposition of the space of paths of a given length into orthogonal subspaces, one of which is the space of essential paths. The space of essential paths can be defined in terms of a representation of the Temperley-Lieb-Jones algebra in the space of paths over the graph. The other terms in the above mentioned decomposition are obtained by means of the application of Ocneanu creation operators to spaces of essential paths of a given length. The product in the resulting weak Hopf algebra is defined using a projection of the concatenation factor by factor of endomorphism of paths. This projection sends graded endomorphism of paths into graded endomorphism of essential paths. The derivation mentioned above can be done for any simple bioriented graph. This provides a generalization of the construction to simple bioriented graphs that are not ADE. In that cases the resulting weak *-Hopf algebra is infinite dimensional. For illustrative purposes a pair of simple examples are considered in this work. One of which is ADE and the other not. Some interesting further research arise in relation to this work. The representation theory of these weak *-Hopf algebras has not been considered in this work. The detailed study of all the affine graphs(β = 2) weak *-Hopf algebras remains to be done. Also the case of non-affine non-ADE graphs(β > 2) is missing. Furthermore the relation of these weak *-Hopf algebras with conformal field theory deserves to be considered. This paper is organized as follows. Sections 2, 3 and 4 set up the scenario and give the basic definitions. Section 5 presents the decomposition and section 6 the projection mentioned above. Sections 7, 8 and 9 deal with the weak Hopf algebra structure. 2. Paths Let G denote a simple biorientable graph. Just to remind the reader some basic definitions to be employed in what follows are included4. Definition 1. Adjacency matrix. Let the graph G have Nv vertices, its adjacency matrix M is the Nv × Nv matrix whose vi vj entry is 1 if the vertex vi is connected to the vertex vj by an edge belonging to G, 0 if it is not connected. Definition 2. Elementary path, length. An elementary path is a succession of consecutive vertices in G. The number of these vertices −1 is called the length of the path. Definition 3. Space of paths P. The inner product vector space of paths P is defined by saying that elementary paths provide a orthonormal basis of this space. 3The question of whether such a derivation exists or not was posed by Oleg Ogievetsky in relation to joint work with the author. 4Further definitions and basic results on graph theory can be found in any textbook on graph theory; a short account of these matters related to this work are presented in appendix A of ref. [9].. Rev. Un. Mat. Argentina, Vol 51-2.

(3) PATHS ON GRAPHS AND ASSOCIATED QUANTUM GROUPOIDS. 149. Definition 4. Concatenation product in P. Given two elementary paths η = (v0 , v1 , · · · , vn ) and η ′ = (v0′ , v1′ , · · · , vn′ ) their concatenation product η ⋆ η ′ is given by, η ⋆ η ′ = δvn v0′ (v0 , v1 , · · · , vn , v1′ , · · · , vn′ ) 3. Creation and annihilation operators on P Let η = (v0 , v1 , · · · , vn ) denote a elementary path of length n. Definition 5. Creation and annihilation operators c†i : Pn → Pn+2 and ci : Pn → Pn−2 ci η. = =. c†i η. = =. ci (v0 , v1 , · · · , vi , vi+1 , vi+2 , vi+3 , · · · , vn ) r µvi+1 (vo , v1 , · · · , vi , vi+3 , · · · , vn ) if 0 ≤ i ≤ n − 2, 0 otherwise δvi vi+2 µvi. c†i (v0 , v1 , · · · , vi , vi+1 , · · · , vn ) X r µv (v0 , v1 , · · · , vi , v, vi , vi+1 , · · · , vn ) if 0 ≤ i ≤ n, 0 otherwise µvi v n.n.v i. (3.1). where Pn is the inner product vector space of paths of length n, µv denotes the components of the Perron-Frobenius eigenvector5 and n.n. denotes nearest neighbours in G. Proposition 6. For i ≤ n,. ci c†i = β1n. where β stands for the highest eigenvalue of the adjacency matrix of G and 1n denotes the identity in the space Pn . Proposition 7. The following operators ei : Pn → Pn , i = 0, · · · , n − 2 , ei =. 1 † c ci β i. give a representation of the Temperley-Lieb-Jones algebra with n − 1 generators. This algebra is defined by the following relations: e2i = ei ,. e†i = ei ,. ei ej = ej ei , |i − j| > 1, ei ei±1 ei =. 1 ei . β2. 5I.e., the eigenvector of the adjacency matrix M with greatest eigenvalue β and with its smallest components taken to be 1.. Rev. Un. Mat. Argentina, Vol 51-2.

(4) 150. R. TRINCHERO. 4. Essential paths Definition 8. Essential Paths Subspace. For each n we denote by En the subspace of Pn defined by the relations ξ ∈ En ⇔ ci ξ = 0, i = 0, · · · , n − 2,. the essential paths subspace E is defined by M E= En . n. This definition implies that. Proposition 9. For all the ADE graphs E is finite dimensional6. In what follows we will denote by {ξa } an orthonormal basis of E (with respect to the restriction to E of the scalar product in P), i.e. (ξa , ξb ) = δab . Example 10. Essentials paths for the graph A3 . The graph A3 , its adjacency matrix, Perron-Frobenius eigenvalue and eigenvector are given below: 0 •. 1 •. . 0 1 M = 1 0 0 1. 2 • ,.    1√ 0 √ 1  , β = 2, µ =  2  . 0 1. Figure 4.1. The graph A3 There are ten essential paths in A3 , which are: • Length zero: (0), (1), (2). • Length one: (01), (12), (10), (21). • Length two: (012), γ = √12 [(121) − (101)], (210). Therefore the maximum length of essential paths over A3 is L = 2. Example 11. Essentials paths for the graph A[2] . The graph A[2] , its adjacency matrix, Perron-Frobenius eigenvalue and eigenvector are given below:. • 0. 1 •. , • 2. . 0 1 M 1 0 1 1.    1 1 1  , β = 2, µ =  1  . 0 1. Figure 4.2. The graph A[2] There are essential paths of any length in A[2] . Any cyclic sequence of consecutive vertices defines an essential path, as for example (120120contiguous120120 · · ·). 6See for example ref.[16] for a proof of this result.. Rev. Un. Mat. Argentina, Vol 51-2.

(5) PATHS ON GRAPHS AND ASSOCIATED QUANTUM GROUPOIDS. 151. 5. Decomposition of the space of paths Definition 12. Maximum length of essential paths. L is the maximum length of essential paths in a graph G iff any path of length greater than L is necessarily non-essential. The following result will be used to write the above mentioned decomposition. Proposition 13. c†i c†j = c†j+2 c†i ∀ j ≥ i (⇒ cj ci = ci cj+2 ∀ j ≥ i). Proof. c†i c†j (v0 , v1 , · · · , vi , · · · ,vj , · · · , vn ) = X r µv † c (v0 , v1 , · · · , vi , · · · , vj , v, vj , · · · , vn ) = µvj i v X r µv µv ′ (v0 , v1 , · · · , vi , v ′ , vi , · · · , vj , v, vj , · · · , vn ), = µ µ v v i j ′ v,v. on the other hand,. c†j+2 c†i (v0 , v1 , · · · , vi , · · · ,vj , · · · , vn ) = X r µv ′ † = cj+2 (v0 , v1 , · · · , vi , v ′ , vi , · · · , vj , · · · , vn ) µ v i v′ X r µv µv ′ (v0 , v1 , · · · , vi , v ′ , vi , · · · , vj , v, vj , · · · , vn ). = µ µ v v i j ′ v,v. . Proposition 14. The operators ci c†j ci c†j ci c†i±1 ci c†i. ci c†j. =. c†j−2 ci. =. c†j ci−2. =. 1n , i ± 1 ≤ n. (5.3). =. β1n , i ≤ n,. (5.4). which imply ci c†j. = =. : Pn → Pn satisfy if i < j − 1. (5.1). if i > j + 1. (5.2). (β δi,j + δi,j+1 + δi,j−1 )1n + θ(j − (i + 2))c†j−2 ci + θ(−j + (i − 2))c†j ci−2. (β δi−j,0 + δi−j,1 + δi−j,−1 )1n + θ(2 − (i − j))c†j−2 ci + θ((i − j) − 2)c†j ci−2 ,. (5.5). where 1n is the identity operator in Pn and the function θ is defined by ( 1 if i ≥ 0 θ(i) = 0 otherwise. Proof. It follows from definition 5..  Rev. Un. Mat. Argentina, Vol 51-2.

(6) 152. R. TRINCHERO. The second equality in (5.5) has been written to emphasize the fact that the coefficients of the different terms depend only on the difference i − j.. Theorem 15. The following decomposition holds7 M † M M Pn = En ci (En−2 ) c†i2 c†i1 (En−4 ) ··· i≤n−2. i1 <i2 ≤n−2. c†i[n/2] c†i[n/2]−1. M. i1 <i2 ···<i[n/2] ≤n−2. · · · c†i1 (E1|0 ),. (5.6). where [ ] denotes the integer part and in the last summand one should take 1 for n odd and 0 for n even. Proof. The following important lemma will be employed in this proof: Lemma 16. For all η ∈ Pn such that ci (η) 6= 0 for some i and cj (η) = 0 ∀ j such that i < j < n − 2 there exist coefficients αk , k = i, · · · , n such that η=. n−2 X. αk c†k (ci (η)) + ξ (i). (5.7). k=i. with ξ (i) satisfying cj (ξ (i) ) = 0 ∀ j such that i − 1 < j < n − 2. Proof. Consider the application of ci to eq.(5.7) ci (η). =. =. n−2 X. k=i n−2 X k=i. αk ci c†k (ci (η)) + ci (ξ (i) ) αk {(β δi,k + δi,k−1 ) + θ(k − (i + 2))c†k−2 ci ](ci (η)) + ci (ξ (i) ). = (βαi + αi+1 )ci (η) + ci (ξ (i) ), where proposition 14 was employed in the second equality and proposition 13 in the third. Therefore if we choose αi and αi+1 such that βαi + αi+1 = 1, then ci (ξ (i) ) = 0. In general the application of ci+l , l = 0, · · · , n − 2 − i to eq. (5.7) is considered: 7Each term in this decomposition can be characterized by the number of non-essential back and forth subpaths. It has certain similarities with what is called Fock’s space in quantum field theory, however they are quite different in some interesting respects. The role of the vacuum is played here by essential paths, so the analogy would be a theory with many non-equivalent vaccums, the number of which could be infinite as for example in the case of A[2] . Excitations are. created out of the vacuum by means of the creation operators c†i . The algebra of these creation and annihilation operators being given by (5.5) which depends on the shape of the graph and which differs significantly from the canonical one, which is given in terms of commutators, appearing in the case of Fock’s space.. Rev. Un. Mat. Argentina, Vol 51-2.

(7) PATHS ON GRAPHS AND ASSOCIATED QUANTUM GROUPOIDS. ci+l (η). =. n−2 X. 153. αk ci+l c†k (ci (η)) + ci+l (ξ (i) ). k=i. =. n−2 X k=i. αk {(β δi+l,k + δi+l,k−1 + δi+l,k+1 ) + θ(2 − (i + l − k))c†k−2 ci+l. +θ(l − 2)θ((i + l − k) − 2))c†k ci+l−2 ](ci (η)) + ci+l (ξ (i) ). = (βαi+l + αi+l+1 + αi+l−1 )ci (η) + ci+l (ξ (i) ). Therefore if the coefficients αk can be chosen such that       . 1 0 0 .. . 0. . . β   1     =    . 1 β 1. 0 1 β .. .. ··· ··· 1 .. . 1.  αi 0  αi+1 0     αi+2   . ..  .   .. β αn−2. .    ,  . then the result follows because ci+l (η) = 0 , l = 1, · · · , n − 2 − i by hypothesis. The determinant of this (n − 1 − i) × (n − 1 − i) matrix can be calculated recursively8 leading to. where.

(8)

(9) β

(10)

(11) 1

(12)

(13) Dn−1−i (β) = det

(14)

(15)

(16)

(17)

(18). 1 β 1. 0 1 β .. .. ··· ··· 1 .. . 1.

(19)

(20)

(21)

(22) n−i n−i

(23) − λ− λ+

(24) ,

(25) = β n−1−i λ+ − λ− . .

(26)

(27) .

(28) β

(29) 0 0. (5.8). p 1 − 4β −2 λ± = . (5.9) 2 For β = 2 the two eigenvalues coincide and taking the limit β → 2 in (5.8) gives 1±. lim Dn−1−i (β) = (n − i),. β→2. which does not vanish for any i(i ≤ n − 2). For β 6= 2, this determinant vanishes if n−i n−i λ+ − λ− =0 ⇒. . λ+ λ−. n−i. = 1 (β 6= 0),. (5.10). 8The same determinant appears in the calculation of the harmonic oscillator transition probability using the path integral (see [17], p. 431).. Rev. Un. Mat. Argentina, Vol 51-2.

(30) 154. R. TRINCHERO. which has no solution for β > 2. For the case β < 2, the ADE case, the eigenvalues are complex conjugate of each other, i.e. λ− = λ∗+ . From eq. (5.9) it is obtained λ+ = eiφ , φ such that β = 2 cos φ, λ−. π where N is the Coxeter number of G. but on the other hand for β < 2, β = 2 cos N However it is well known that the maximum length of essential paths L is related π π = L+2 , therefore eq. (5.10) is to the Coxeter number by L = N − 2, thus φ = N the same as. ei. π(n−i) L+2. = 1,. which can never be satisfied. This is so because under the assumptions of this lemma the following inequality should hold n − i − 1 ≤ L. If it were not so then the path obtained by reversing η and including the first n − i − 1 steps would be an essential path of length greater than L in the graph G which is impossible by definition of L.  Using this lemma the following algorithm can be employed to obtain a unique decomposition of an arbitrary path η ∈ Pn as in the r.h.s. of (5.6). Decompose η as in (5.7). Then decompose every ci (η) appearing in the first term of the r.h.s. of (5.7) using (5.7) and do the same with ξ (i) . At each step of this process the resulting paths are annihilated by one more ci operator, since the number of these operators that can act on an element of Pn is n − 2 then this process necessarily converges to something belonging to the r.h.s. of (5.6). The ordering of the indices of the c† operators in (5.6) follows using proposition 13.  The following result is a simple consequence of the decomposition (5.6). Proposition 17. The subspaces of Pn given by M Pn(l) = c†i1 c†i2 · · · c†il (En−2l ) , P (0) (n) = En , l = 0, · · · , [n/2], i1 <i2 <···≤n−2. are mutually orthogonal. Proof. This proposition is proved if we show that Mlm = (c†i1 c†i2 · · · c†il (ξ (n−2l) ), c†j1 c†j2 · · · c†jm (ξ (n−2m) )) ∝ δlm ,. ∀ξ (n−2l) ∈ En−2l , ξ (n−2m) ∈ En−2m ,. this in turn follows9 from the relations in proposition 14.. . Thus there exist orthogonal projections on each of the subspaces P (l) , l = 0, (l) (l) (l) (l) · · · , [n/2] that we denote by Πn and satisfy Πn = Πn 2 = Πn † . In particular (0) Πn is a orthogonal projector over essential paths of length n. 9See the proof of proposition 20 for a similar argument.. Rev. Un. Mat. Argentina, Vol 51-2.

(31) PATHS ON GRAPHS AND ASSOCIATED QUANTUM GROUPOIDS. 155. Example 18. Decomposition of non-essential paths in A3 . Using the algorithm of the previous theorem the following decompositions of non-essential paths of a given length coming from the concatenation of essential paths are obtained: Length two: (01) ⋆ (10) = (010) = (21) ⋆ (12) = (212) = (10) ⋆ (01) = (101) = (12) ⋆ (21) = (121) =. 1 † c (0) 21/4 0 1 † c (2) 21/4 0 1 1 √ ( 1/4 c†0 (1) − γ) 2 2 1 1 √ ( 1/4 c†0 (1) + γ) 2 2. Length three10: 1 † c1 1/4 2. (01) ⋆ γ. =. (. (21) ⋆ γ. =. −(. 1 † c1 1/4 2. 1 † c0 1/4 2. γ ⋆ (10) =. (. γ ⋆ (12) =. −(. − 21/4 c†0 )(01) − 21/4 c†0 )(21). − 21/4 c†1 )(10). 1 † c0 1/4 2. − 21/4 c†1 )(12). (10) ⋆ (012) =. (21/4 c†0 −. 1 † c1 )(12) 1/4 2. (012) ⋆ (21) =. (21/4 c†1 −. 1 † c0 )(01) 1/4 2. (12) ⋆ (210) =. (21/4 c†0 −. 1 † c1 )(10) 1/4 2. (210) ⋆ (01) =. (21/4 c†1 −. 1 † c0 )(21) 1/4 2. Length four: 1 (012) ⋆ (210) = (c†1 − √ c†2 ) c†0 ((0)) 2 1 (210) ⋆ (012) = (c†1 − √ c†2 ) c†0 ((2)) 2 1 † † † γ ⋆ γ = (c1 − √ c2 )c0 ((1)) 2. (5.11) (5.12) (5.13). 10It is recalled that γ = √1 [(121) − (101)] as defined in example 10. 2. Rev. Un. Mat. Argentina, Vol 51-2.

(32) 156. R. TRINCHERO. Example 19. Decomposition of non-essential paths in A[2] . Using the algorithm of the previous theorem the following decompositions of non-essential paths of a given length coming from the concatenation of essential paths are obtained: Length two: 1 † c ((0)) + 2 0 1 † (12) ⋆ (21) = c0 ((1)) + 2 1 † (20) ⋆ (02) = c0 ((2)) + 2 (01) ⋆ (10) =. 1 ξ010 , 2 1 ξ121 , 2 1 ξ202 , 2. (02) ⋆ (20) =. 1 † 2 c0 ((0)). −. 1 2 ξ020 ,. ξ010 = ξ020 = (010) − (020) ∈ E,. (10) ⋆ (01) =. 1 † 2 c0 ((1)). −. 1 2 ξ101 ,. ξ121 = ξ101 = (121) − (101) ∈ E,. (21) ⋆ (12) =. 1 † 2 c0 ((2)). −. 1 2 ξ212 ,. ξ202 = ξ212 = (202) − (212) ∈ E.. It is worth noting that the last two lines above can be obtained from the first one by making cyclic permutations of the vertices 0, 1 and 2 (not for the indices of the c† operators), i.e. by applying the rotations contained in the symmetry group C3v of the graph A[2] . Length three: 2 1 (10) ⋆ (012) = (1012) = ( c†0 − c†1 )(12) + ξ1012 3 3 2 † 1 † (012) ⋆ (21) = (0121) = ( c1 − c0 )(01) + ξ0121 3 3 2 † 1 † (10) ⋆ ξ010 = (1010) − (1020) = ( c0 − c1 )(10) + ξ(10)⋆ξ010 3 3 2 † 1 † ξ010 ⋆ (01) = (0101) − (0201) = ( c1 − c0 )(01) + ξξ010 ⋆(01) 3 3 where 1 [(1012) − (1212) + (1202)] ∈ E 3 1 [(0121) − (0101) + (0201)] ∈ E ξ0121 = 3 2 ξ(10)⋆ξ010 = [(1010) − (1020) − (1210)] ∈ E 3 2 ξξ010 ⋆(01) = [(0101) − (0201) − (0121)] ∈ E 3 From these four decompositions and applying the elements of the symmetry group C3v of the graph A[2] the other twenty decompositions can be readily obtained. Length four: ξ1012. =. 2 1 1 1 1 1 (2) (2) (0) (01210) = ( c†1 − c†2 ) [ c†0 ((0)) + ξ01210 ] − ( c†0 − c†1 + c†2 )ξ01210 + ξ01210 (5.14) 3 3 2 2 3 6 where ξ (0) , ξ (2) ∈ E are given by, 1 (0) [(01210) + (02120) + (01020) − (02020) − (01010) + (02010)] ξ01210 = 6 1 (2) ξ01210 = [(010) − (020)]. 2 Rev. Un. Mat. Argentina, Vol 51-2.

(33) 157. PATHS ON GRAPHS AND ASSOCIATED QUANTUM GROUPOIDS. Also 1 ξ010 ⋆ (012) = [c†1 − (c†2 + c†0 )](012) + ξξ010 ⋆(012) 2 1 (210) ⋆ ξ010 = [c†1 − (c†2 + c†0 )](210) + ξ(210)⋆ξ010 , 2 where ξξ010 ⋆(012) , ξ(210)⋆ξ010 ∈ E are given by ξξ010 ⋆(012). =. ξ(210)⋆ξ010. =. 1 [(01012) − (02012) − (01212) + (01202)] 2 1 [(21010) − (21020) − (21210) + (20210)]. 2. Applying the elements of C3v to these decompositions the others can be readily obtained. Finally, ξ121 ⋆ ξ121 =. 2 † † 1 c c (1) − c†2 c†1 (1) + ξξ121 ⋆ξ121 , 3 1 1 3. where ξξ121 ⋆ξ121 ∈ E is given by ξξ121 ⋆ξ121 =. 2 [(12121) + (10101) − (10121) − (12101) − (12021) − (10201)]. 3 6. The projection. A posteriori motivation for the definition of the projection P : Endgr (P) → Endgr (E) appearing below is given by its properties with respect to the concatenation product of paths (see propositions 27, 29, 36, 37, 39, 43). However it was proposed based on its relation with the representation theory of these weak *-Hopf algebras, representation theory that is not considered in this paper. Before giving this definition a useful result is given: Proposition 20. (ci1 ci2 · · · cin c†jn c†jn−1 · · · c†j1 ξa , ξb ) = δab C(i1 , · · · ; in ; jn , · · · ; j1 ),. (6.1). where ξa , ξb are elements of an orthonormal basis of E such that #ξa = #ξb ≥ j1 and #ξa = #ξb ≥ i1 . Proof. The evaluation of the matrix element in the l.h.s. is considered. By means of relation (5.5) the product of operators cin c†jn is either replaced by a number β or 1 (which we call a contraction), or they are interchanged with a change in the index of one of them. In any case for the matrix element to be non-vanishing, all the i indices should be contracted with j indices. If this is not the case the matrix element vanishes because necessarily a c operator will be applied to ξa or ξb which gives zero because they are essential.  Rev. Un. Mat. Argentina, Vol 51-2.

(34) 158. R. TRINCHERO. Let End(Pn ) (End(En )) denote the vector space of endomorphism of length n paths (essential paths)11. In what follows the vector space of length preserving endomorphism of paths (essential paths) will be considered. They are defined by M M Endgr (P) = End(Pn ) , Endgr (E) = End(En ). n 12. n. gr. gr. Definition 21. A projector P : End (P) → End (E) is defined by its action on the terms appearing in the decomposition (5.6) X P (c†jn · · · c†j1 ξa ⊗ c†in · · · c†i1 ξb ) = (ci1 ci2 · · · cin c†jn c†jn−1 · · · c†j1 ξa , ξc ) ξc ⊗ ξb =. ξc ∈E. C(i1 , · · · , in ; jn , · · · , j1 ) ξa ⊗ ξb ,. (6.2). where j1 < j2 < · · · < jn and i1 < i2 < · · · < in .. It is clear that P 2 = P but P † 6= P which implies that P is not an orthogonal projection.. Remark 22. It should be noted that the projection of an arbitrary element η ⊗ η ′ of End(Pn ) is obtained by applying definition 21 to each term appearing in the decomposition of η ⊗ η ′ as in eq. (5.6). Thus in general this projection consists in a summation of elements belonging to [n/2]. M. End(En−2l ).. l=0. Therefore it will not in general respect the grading. Remark 23. Note that because of eq. (6.1) and the orthonormality of the basis {ξa }, the following equality holds =. X. ξc ∈E. P (c†jn c†jn−1 · · · c†j1 ξa ⊗ c†in c†in−1 · · · c†i1 ξb ) =. ξa ⊗ ξc (ξc , cj1 cj2 · · · cjn c†in c†in−1 · · · c†i1 ξb ).. Example 24. As an example the projections of the element (01210) ⊗ (21012) both for the graph A3 and A{2} are calculated. For A3 : X † 1 1 P ((01210) ⊗ (21012))A3 = ((c1 − √ c†2 ) c†0 ((0)), (c†1 − √ c†2 ) c†0 ((v))) v ⊗ (2) 2 2 v X 1 1 = (c0 (c1 − √ c2 ) (c†1 − √ c†2 ) c†0 ((0)), v) v ⊗ (2) 2 2 v √ X√ √ 1 2 1 ) δv,0 v ⊗ (2) = 0 ⊗ 2, = 2( 2 − √ − √ + 2 2 2 v 11In what follows the following equalities End(P ) = P ⊗ P and End(E ) = E ⊗ E will n n n n n n be employed. This is so because by means of the scalar product appearing in definition 3 and section 8 it is possible to identify Pn and En with their duals. 12In ref. [18] another projector Q acting on the same vector space is considered. Defining a product as in (7.3) but using Q does not lead to a weak Hopf algebra structure.. Rev. Un. Mat. Argentina, Vol 51-2.

(35) 159. PATHS ON GRAPHS AND ASSOCIATED QUANTUM GROUPOIDS. where the decompositions (5.11) and (5.12) were employed for the first equality and proposition 14 for the last. For A[2] , the decomposition of eq. (5.14) and the following are employed: 2 1 1 1 1 1 (2) (2) (0) (21012) = ( c†1 − c†2 ) [ c†0 ((2)) + ξ21012 ] − ( c†0 − c†1 + c†2 )ξ21012 + ξ21012 , 3 3 2 2 3 6 where 1 (2) [(212) − (202)], ξ21012 = 2 (1) ξ21012 = 61 [ (21012) + (20102) + (21202) − (20202) − (21212) + (20212)]. Recalling that the projection kills terms with unequal number of c† operators applied to essential paths in each factor of the tensor product leads to X (1) (1) P ((01210) ⊗ (21012))A[2] = (ξ01210 , ρ) ρ ⊗ ξ21012 ρ∈E. X † 1 † 1 (2) (2) + (([c1 − (c0 + c†2 )]ξ01210 , [c†1 − (c†0 + c†2 )]ρ) ρ ⊗ ξ21012 2 2 ρ∈E. +. X. v∈E0. 2 1 1 2 1 1 (( c†1 − c†2 ) c†0 (0), ( c†1 − c†2 ) c†0 (v)) v ⊗ (2). 3 3 2 3 3 2. Evaluating the scalar products gives (1). (1). (2). (2). P ((01210) ⊗ (21012))A[2] = ξ01210 ⊗ ξ21012 + ξ01210 ⊗ ξ21012 +. 1 (0) ⊗ (2). 3. 7. Star algebra In the vector space Endgr (P) the following involution is considered: Definition 25. Star. (ξ ⊗ ξ ′ )⋆ = ξ ⋆ ⊗ ξ ′⋆ , (7.1) ⋆ where ξ denotes the path obtained from ξ by "time inversion" for elementary paths and extending antilinearly to all P, i.e., by reversing the sense in which the succession of contiguous vertices is followed for elementary paths, i.e., ξ = (vo , v1 , · · · , vn−1 , vn ) ⇒ ξ ⋆ = (vn , vn−1 , · · · , v1 , v0 ). From this definition and the one of the scalar product in section 2, it is clear that (η, χ) = (η ⋆ , χ⋆ ),. (7.2). where the bar indicates the complex conjugate. The underlying vector space of the algebra to be considered is given by the length graded endomorphisms of essential paths13 Endgr (E). The product is defined by: 13This choice of the underlying vector space structure doe not mean that the product to be considered is the composition of endomorphisms in Endgr (E). It is emphasized that this is not the product to be considered but another product that we call · and that will be defined below.. Rev. Un. Mat. Argentina, Vol 51-2.

(36) 160. R. TRINCHERO. Definition 26. Product. (ξ ⊗ ξ ′ ) · (ρ ⊗ ρ′ ) = P (ξ ⋆ ρ ⊗ ξ ′ ⋆ ρ′ ) ; ξ, ρ, ξ ′ , ρ′ ∈ E.. (7.3). This product does not make this algebra a graded one. This is a filtered algebra respect to the length of paths. The product of ξ ⊗ ξ ′ ∈ End(En1 ) with ρ ⊗ ρ′ ∈ End(En2 ) in general belongs to [(n1 +n2 )/2]. M l=0. End(En1 +n2 −2l ).. The identity is 1=. X. v,v ′ ∈E0. v ⊗ v′ .. The properties to be fulfilled by these definitions are fairly simple to be proved except for the antihomomorphism property of the involution (7.1) and the associativity of the product (7.3). The following result will be employed in the proof of the first of these properties: Proposition 27. P ((η ⊗ η ′ )⋆ ) = P (η ⊗ η ′ )⋆ ∀η, η ′ ∈ P.. (7.4). Proof. Typical contributions to the decomposition (5.6) for η and η ′ are considered. These contributions should have the same number of c† operators applied to essential paths in order to have a non-vanishing image when applying the projector. Therefore the following expression is considered P (c†jn c†jn−1 · · · c†j1 ξa ⊗ c†in c†in−1 · · · c†i1 ξb ) = X (ci1 ci2 · · · cin c†jn c†jn−1 · · · c†j1 ξa , ξc ) ξc ⊗ ξb ξc ∈E. =. X. ξc ∈E. (c†jn c†jn−1 · · · c†j1 ξa , c†in c†in−1 · · · c†i1 ξc ) ξc ⊗ ξb ,. thus P (c†jn c†jn−1 · · · c†j1 ξa ⊗c†in c†in−1 · · · c†i1 ξb )⋆ =. X. (c†jn c†jn−1 · · · c†j1 ξa , c†in c†in−1 · · · c†i1 ξc ) ξc⋆ ⊗ξb⋆ .. ξc ∈E. The time inversion of a path of the form c†in c†in−1 · · · c†i1 ξb leads to (the counting starts from the end of this path) (c†in · · · c†i2 c†i1 ξb )⋆ = c†l+2(n−1)−in c†l+2(n−2)−in−1 · · · c†l+2−i2 c†l−i1 (ξb⋆ ), Rev. Un. Mat. Argentina, Vol 51-2.

(37) 161. PATHS ON GRAPHS AND ASSOCIATED QUANTUM GROUPOIDS. where l = #ξb . Using this relation leads to P ((c†jn c†jn−1 · · · c†j1 ξa )⋆ ⊗ (c†in c†in−1 · · · c†i1 ξb )⋆ ) =. = =. P. † † ξd⋆ ∈E ((cjn cjn−1. † † ξd⋆ ∈E ((cjn cjn−1. P. P. =. · · · c†j1 ξa )⋆ , c†l+2(n−1)−in · · · c†l+2−i2 c†l−i1 ξd⋆ ) ξd⋆ ⊗ ξb⋆. ξd ∈E. · · · c†j1 ξa )⋆ , (c†in c†in−1 · · · c†i1 ξd )⋆ ) ξd⋆ ⊗ ξb⋆. (c†jn c†jn−1 · · · c†j1 ξa , c†in c†in−1 · · · c†i1 ξd ) ξd⋆ ⊗ ξb⋆ ,. where in the last equality eq. (7.2) was employed and the fact that when ξd runs over all E then ξd⋆ also.  Using (7.4) it follows that Proposition 28. ((ξ ⊗ ξ ′ ) · (ρ ⊗ ρ′ ))⋆ = (ρ ⊗ ρ′ )⋆ · (ξ ⊗ ξ ′ )⋆ ∀ξ, ξ ′ , ρ, ρ′ ∈ E.. (7.5). Proof. (ρ ⊗ ρ′ )⋆ · (ξ ⊗ ξ ′ )⋆. = P ((ρ ⊗ ρ′ )⋆ ⋆ (ξ ⊗ ξ ′ )⋆ ) = P (((ξ ⊗ ξ ′ ) ⋆ (ρ ⊗ ρ′ ))⋆ ) = P ((ξ ⊗ ξ ′ ) ⋆ (ρ ⊗ ρ′ ))⋆ = ((ξ ⊗ ξ ′ ) · (ρ ⊗ ρ′ ))⋆ . . In order to prove the associativity of the product (7.3) the following preliminary result is considered Proposition 29. P ((ξ ⊗ ξ ′ ) ⋆ P (η ⊗ η ′ )) P (P (η ⊗ η ′ ) ⋆ (ξ ⊗ ξ ′ )). = P ((ξ ⊗ ξ ′ ) ⋆ (η ⊗ η ′ )) (7.6) ′ ′ ′ ′ = P ((η ⊗ η ) ⋆ (ξ ⊗ ξ )) ∀ξ, ξ ∈ E, η, η ∈ P. (7.7). Proof. As in proposition 27, typical contributions to the decomposition of η ⊗ η ′ are considered. Thus the following expression is dealt with P ((ξ ⊗ ξ ′ ) ⋆ P (c†jn c†jn−1 · · · c†j1 ξa ⊗ c†in c†in−1 · · · c†i1 ξb )) =. = C(i1 , · · · , in ; jn , · · · , j1 ) P ((ξ ⊗ ξ ′ ) ⋆ (ξa ⊗ ξb )),. where it was assumed that P (c†in c†in−1 · · · c†i1 ξa ⊗ c†jn c†jn−1 · · · c†j1 ξb ) does not vanish (if it vanishes it can be easily seen that the r.h.s. of eq.(7.6) also vanishes). Next the r.h.s. of eq.(7.6) is considered P ((ξ ⊗ ξ ′ ) ⋆ (c†jn c†jn−1 · · · c†j1 ξa ⊗ c†in c†in−1 · · · c†i1 ξb )) = = = =. P (ξ ⋆ c†jn c†jn−1 · · · c†j1 ξa ⊗ ξ ′ ⋆ c†in c†in−1 · · · c†i1 ξb ). P (c†l+jn · · · c†l+j1 (ξ ⋆ ξa ) ⊗ c†l+in · · · c†l+i1 (ξ ′ ⋆ ξb )). C(l + i1 · · · , l + in ; l + jn , · · · , l + j1 )P ((ξ ⊗ ξ ′ ) ⋆ (ξa ⊗ ξb )),. where l denotes the length of the path ξ. From its definition (6.1) it follows that C(i1 , · · · , in ; jn , · · · , j1 ) = C(l + i1 · · · , l + in ; l + jn , · · · , l + j1 ) Rev. Un. Mat. Argentina, Vol 51-2.

(38) 162. R. TRINCHERO. which completes the proof of the first equality. Eq.(7.7) follows along identical lines.  Using this result associativity follows Proposition 30. ((ξ1 ⊗ ξ1′ )·(ξ2 ⊗ ξ2′ ))·(ξ3 ⊗ ξ3′ ) = (ξ1 ⊗ ξ1′ )·((ξ2 ⊗ ξ2′ )·(ξ3 ⊗ ξ3′ )) ∀ξi , ξi′ ∈ E , i = 1, 2, 3. Proof.. ((ξ1 ⊗ ξ1′ ) · (ξ2 ⊗ ξ2′ )) · (ξ3 ⊗ ξ3′ ). = P (P ((ξ1 ⊗ ξ1′ ) ⋆ (ξ2 ⊗ ξ2′ )) ⋆ (ξ3 ⊗ ξ3′ )) = P ((ξ1 ⊗ ξ1′ ) ⋆ (ξ2 ⊗ ξ2′ ) ⋆ (ξ3 ⊗ ξ3′ )). = P ((ξ1 ⊗ ξ1′ ) ⋆ P ((ξ2 ⊗ ξ2′ ) ⋆ (ξ3 ⊗ ξ3′ ))) = (ξ1 ⊗ ξ1′ ) · ((ξ2 ⊗ ξ2′ ) · (ξ3 ⊗ ξ3′ )).. . Example 31. Product for the case of A3 . It can be explicitly verified that in this case the product coincides with the one of the double triangle algebra14 described in ref.[11]. For illustrative purposes the calculation of some of these products is given below:   1 † 1 † 1 c (2) ⊗ ( c (1) + γ) (21 ⊗ 12) · (12 ⊗ 21) = P (212 ⊗ 101) = √ P 21/4 0 21/4 0 2 1 = √ (2) ⊗ (1) 2   1 1 † 1 † (10 ⊗ 12) · (01 ⊗ 21) = P ( 1/4 c0 (1) − γ) ⊗ ( 1/4 c0 (1) + γ) 2 2 2 1 = (1 ⊗ 1 − γ ⊗ γ) 2   1 † 1 † 1 (12 ⊗ 12) · (21 ⊗ 21) = P ( 1/4 c0 (1) + γ) ⊗ ( 1/4 c0 (1) + γ) 2 2 2 1 = (1 ⊗ 1 + γ ⊗ γ) 2 (γ ⊗ 012) · (10 ⊗ 21) = P (γ ⋆ 10 ⊗ 0121)   1 † 1 † 1/4 † 1/4 † = P ( 1/4 c0 − 2 c1 )(10) ⊗ (2 c1 − 1/4 c0 )(01) 2 2 = −(10) ⊗ (01)   1 1 (γ ⊗ γ) · (γ ⊗ γ) = P (c†1 − √ c†2 )c†0 (1) ⊗ (c†1 − √ c†2 )c†0 (1) = (1) ⊗ (1). 2 2 14In that reference product is calculated using the pairing with the dual algebra, i.e. the 6j-symbols using the dual product (that is the composition of endomorphisms) and coming back with the 6j-symbols. The same construction for the case of A[2] is not known. The extension of 6j-symbols (Ocneanu cells) for this last case is not obvious since for example A[2] is not a bicolorable graph.. Rev. Un. Mat. Argentina, Vol 51-2.

(39) PATHS ON GRAPHS AND ASSOCIATED QUANTUM GROUPOIDS. 163. Example 32. Product for the case of A[2] . With the notation of example 19 the following illustrative products can be computed: (21 ⊗ 12) · (12 ⊗ 21) = P (212 ⊗ 121)   1 1 1 1 1 1 † c0 (2) − ξ212 ⊗ c†0 (1) + ξ121 = (2) ⊗ (1) − ξ212 ⊗ ξ121 =P 2 2 2 2 2 4 (10 ⊗ 12) · (01 ⊗ 21) = P (101 ⊗ 121)   1 1 1 1 = P ( c†0 (1) − ξ101 ) ⊗ ( c†0 (1) + ξ121 ) 2 2 2 2 1 1 = (1) ⊗ (1) − ξ121 ⊗ ξ121 ) 2 4 (12 ⊗ 12) · (21 ⊗ 21) = P (121 ⊗ 121)   1 1 1 1 = P ( c†0 (1) + ξ121 ) ⊗ ( c†0 (1) + ξ121 ) 2 2 2 2 1 1 = (1) ⊗ (1) + ξ121 ⊗ ξ121 ) 2 4 (ξ121 ⊗ 012) · (10 ⊗ 21) = P (ξ121 ⋆ 10 ⊗ 0121)   2 † 1 † 2 † 1 † = P ( c1 − c0 )(10) + ξξ121 ⋆10 ⊗ ( c1 − c0 )(01) + ξ0121 3 3 3 3 2 = (10) ⊗ (01) + ξξ121 ⋆10 ⊗ ξ0121 3 (ξ121 ⊗ ξ121 ) · (ξ121 ⊗ ξ121 ) =   2 † † 1 † † 2 † † 1 † † = P ( c1 c1 − c2 c1 )(1) + ξξ121 ⋆ξ121 ⊗ ( c1 c1 − c2 c1 )(1) + ξξ121 ⋆ξ121 3 3 3 3 4 = (1) ⊗ (1) + ξξ121 ⋆ξ121 ⊗ ξξ121 ⋆ξ121 . 3. 8. Weak bialgebra The definition of a weak ⋆-bialgebra is recalled Definition 33. A weak ⋆-bialgebra is a ⋆-algebra A together with two linear maps ∆ : A → A ⊗ A, the coproduct, and ǫ : A→ C, the counit, satisfying the following axioms ∆(ab) = ⋆. ∆(a ) = (∆ ⊗ Id)∆. =. ∆(a)∆(b) ∆(a)⋆ (Id ⊗ ∆)∆, Rev. Un. Mat. Argentina, Vol 51-2.

(40) 164. R. TRINCHERO. and ǫ(ab). =. ǫ(a11 )ǫ(12 b). (ǫ ⊗ Id)∆ = Id ǫ(aa⋆ ) ≥. = (Id ⊗ ǫ)∆ 0,. where in the first equation Sweedler convention is employed and also in the following equation that defines 11 and 12 ∆(1) = 11 ⊗ 12 , with 1 being the identity in A. The definition of coproduct and counit considered for the star algebra of the previous section are Definition 34. Coproduct15, ∆(ξ ⊗ ξ ′ ) =. X. ξa ∈E. ξ ⊗ ξa ⊠ ξ a ⊗ ξ ′ ,. (8.1). #ξa =#ξ. where the summation runs over a complete orthonormal basis for E. Definition 35. Counit, ǫ(ξ ⊗ ξ ′ ) = (ξ, ξ ′ ). The axioms appearing in the definition of a weak bialgebra are fairly simple to prove for the above definitions except for the morphism property for the coproduct and the one involving the counit of a product. For the first property the following preliminary results are useful: Proposition 36. ∆P = P ⊗2 ∆P , where ∆P (χ ⊗ χ′ ) = thonormal basis of P.. P. η∈P. (8.2). χ ⊗ η ⊠ η ⊗ χ′ with summation over a complete or-. Proof. Eq. (8.1) is applied to a generic element η ⊗ η ′ of End(P), typical terms in the decomposition (5.6) with non-vanishing image by the projector are considered ∆P (c†jn c†jn−1 · · · c†j1 ξa ⊗ c†in c†in−1 · · · c†i1 ξb ) =. X. C(i1 , · · · , in ; jn , · · · , j1 ) ξa ⊗ ξc ⊠ ξc ⊗ ξb ,. ξc ∈E.. 15In the dual weak Hopf algebra to the one considered here, this coproduct maps to the product. Eq. (8.1) implies that this product corresponds to the composition of endomorphisms in the dual weak Hopf algebra.. Rev. Un. Mat. Argentina, Vol 51-2.

(41) PATHS ON GRAPHS AND ASSOCIATED QUANTUM GROUPOIDS. 165. where it was assumed that P (c†jn c†jn−1 · · · c†j1 ξ ⊗ c†in c†in−1 · · · c†i1 ξ ′ ) does not vanish. On the other hand = = = = = =. P ⊗2 ∆P (c†jn c†jn−1 · · · c†j1 ξa ⊗ c†in c†in−1 · · · c†i1 ξb ) = P P ⊗2 ( η∈P c†jn c†jn−1 · · · c†j1 ξa ⊗ η ⊠ η ⊗ c†in c†in−1 · · · c†i1 ξb ) P P ξc ,ξd ∈E η∈P (ξc , cj1 · · · cjn η)(ci1 · · · cin η, ξd ) ξa ⊗ ξc ⊠ ξd ⊗ ξb P P † † † † ξc ,ξd ∈E η∈P (cjn · · · cj1 ξc , η)(η, cin · · · ci1 ξd ) ξa ⊗ ξc ⊠ ξd ⊗ ξb P † † † † ξc ,ξd ∈E (cjn · · · cj1 ξc , cin · · · ci1 ξd ) ξa ⊗ ξc ⊠ ξd ⊗ ξb P † † ξ ,ξ ∈E (ξc , cj1 · · · cjn cin · · · ci1 ξd ) ξa ⊗ ξc ⊠ ξd ⊗ ξb Pc d ξc ∈E C(i1 , · · · , in ; jn , · · · , j1 ) ξa ⊗ ξc ⊠ ξc ⊗ ξb .. . Proposition 37. P ⊗2 (∆P (ξa ⊗ ξb ) ⋆ ∆P (ξc ⊗ ξd )) = P ⊗2 [P ⊗2 ∆P (ξa ⊗ ξb ) ⋆ P ⊗2 ∆P (ξc ⊗ ξd )]. Proof. P ⊗2 (∆P (ξa ⊗ ξb ) ⋆ ∆P (ξc ⊗ ξd )) = X = P ⊗2 ( (ξa ⊗ η ⊠ η ⊗ ξb ) ⋆ (ξc ⊗ χ ⊠ χ ⊗ ξd )) η,χ∈P. =P. ⊗2. (. X. η,χ∈P. On the other hand,. P (ξa ⋆ ξc ⊗ η ⋆ χ) ⊠ P (η ⋆ χ ⊗ ξb ⋆ ξd )).. P ⊗2 [P ⊗2 ∆P (ξ ⊗ ξ ′ ) ⋆ P ⊗2 ∆P (ρ ⊗ ρ′ )] = X = P ⊗2 [P ⊗2 (ξa ⊗ η ⊠ η ⊗ ξb ) ⋆ P ⊗2 ∆P (ξc ⊗ χ ⊠ χ ⊗ ξd )] η,χ∈P. = P ⊗2. X. η,χ∈P. P (P (ξa ⊗ η) ⋆ P (ξc ⊗ χ)) ⊠ P (P (η ⊗ ξb ) ⋆ P (χ ⊗ ξd )). (8.3). Next it is noted that P (ξa ⊗ η) ⋆ P (ξc ⊗ χ) = =. X. ω,σ∈E. X. ω,σ∈E. = Replacing in (8.3) leads to. (ω, η)(σ, χ)(ξa ⊗ ω) ⋆ (ξc ⊗ σ) (ξa ⋆ ξc ) ⊗ (ω ⋆ σ) δωη δσχ. (ξa ⋆ ξc ) ⊗ (η ⋆ χ).. P ⊗2 [P ⊗2 ∆P (ξa ⊗ξb )⋆P ⊗2 ∆P (ξc ⊗ξd )] = P ⊗2 (. X. η,χ∈P. P (ξa ⋆ξc ⊗η⋆χ)⊠P (η⋆χ⊗ξb ⋆ξd )).  Rev. Un. Mat. Argentina, Vol 51-2.

(42) 166. R. TRINCHERO. Using the above results leads to Proposition 38. ∆((ξ ⊗ ξ ′ ) · (ρ ⊗ ρ′ )) = ∆(ξ ⊗ ξ ′ ) · ∆(ρ ⊗ ρ′ ).. Proof.. ∆((ξ ⊗ ξ ′ ) · (ρ ⊗ ρ′ )). = ∆(P ((ξ ⊗ ξ ′ ) ⋆ (ρ ⊗ ρ′ ))). = P ⊗2 ∆P ((ξ ⊗ ξ ′ ) ⋆ (ρ ⊗ ρ′ )). = P ⊗2 (∆P (ξ ⊗ ξ ′ ) ⋆ ∆P (ρ ⊗ ρ′ )). = P ⊗2 [P ⊗2 ∆P (ξ ⊗ ξ ′ ) ⋆ P ⊗2 ∆P (ρ ⊗ ρ′ )] = P ⊗2 [∆(P (ξ ⊗ ξ ′ )) ⋆ ∆(P (ρ ⊗ ρ′ ))]. = ∆(ξ ⊗ ξ ′ ) · ∆(ρ ⊗ ρ′ ).. . Regarding the counit of a product the following result will be employed: Proposition 39. ǫ(P (η ⊗ η ′ )) = ǫ(η ⊗ η ′ ) ∀ η, η ′ ∈ P.. Proof. A generic element η ⊗ η ′ of End(P) is considered, typical terms in the decomposition (5.6) with non-vanishing image by the projector are considered: ǫ(P (c†jn c†jn−1 · · · c†j1 ξa ⊗ c†in c†in−1 · · · c†i1 ξb )) = X = ǫ( (ci1 ci2 · · · cin c†jn c†jn−1 · · · c†j1 ξa , ξc ) ξc ⊗ ξb ) ξc ∈E. = (ci1 ci2 · · · cin c†jn c†jn−1 · · · c†j1 ξa , ξb ). = ǫ(c†jn c†jn−1 · · · c†j1 ξa ⊗ c†in c†in−1 · · · c†i1 ξb ). . Thus, Proposition 40. Proof.. ǫ((ξa ⊗ ξb ) · (ξc ⊗ ξd )) = ǫ((ξa ⊗ ξb ) · 11 )ǫ(12 · (ξc ⊗ ξd )).. ǫ((ξa ⊗ ξb ) · (ξc ⊗ ξd )) = ǫ(P (ξa ⋆ ξc ⊗ ξb ⋆ ξd )) = ǫ(ξa ⋆ ξc ⊗ ξb ⋆ ξd ) = (ξa ⋆ ξc , ξb ⋆ ξd ). On the other hand,. ǫ((ξa ⊗ ξb ) · 11 )ǫ(12 · (ξc ⊗ ξd )) = =. X. v,u,v ′. X. ǫ((ξa ⊗ ξb ) · (v ⊗ u))ǫ((u ⊗ v ′ ) · (ξc ⊗ ξd )) δr(ξa )v δr(ξb )u δus(ξc ) δv′ s(ξd ) (ξa , ξb )(ξc , ξd ). v,u,v ′. =. Rev. Un. Mat. Argentina, Vol 51-2. (ξa ⋆ ξc , ξb ⋆ ξd ) = ǫ((ξa ⊗ ξb ) · (ξc ⊗ ξd )).. .

(43) PATHS ON GRAPHS AND ASSOCIATED QUANTUM GROUPOIDS. 167. 9. The antipode In general the axioms to be satisfied by the antipode are S(ab) = S((S(a⋆ )⋆ ) = ∆(S(a)) S(a1 ) · a2 ⊗ a3. = =. S(b)S(a) a S ⊗ S (∆op (a)) 11 ⊗ a12 ,. (9.1). where in the last equation Sweedler convention has been employed. The following ansatz for the antipode is considered: S(ξ ⊗ ω) = F (ξ, ω) ω ⋆ ⊗ ξ ⋆ ,. (9.2). where F (ξ, ω) is a numerical factor to be determined. It is fairly simple to show that the first three axioms in (9.1) are satisfied by this definition. The proof of the last axiom is more involved. The following preliminary results are considered: Proposition 41. The following holds, c†n−1 c†n−2. · · · c†0 (v0 ). =. X. η∈Pn /s(η)=v0. s. µr(η) η ⋆ η⋆ , µs(η). (9.3). where the summation is over the orthonormal basis of elementary paths with starting vertex v0 and, s X µr(ξ) † † (0) (0) † ξ ⋆ ξ⋆, (9.4) (Πn ⋆ Πn )cn−1 cn−2 · · · c0 (v0 ) = µs(ξ) ξ∈En /s(η)=v0. (0). where Πn is the orthogonal projector over essential paths of length n mentioned (0) (0) after proposition 17. The notation (Πn ⋆ Πn ) indicates that when applied to a path of length 2n this operator projects over paths that are essential in its first n steps and also essential in its last n steps. Proof. It follows from definition (3.1).  Proposition 42. Let ξ, ρ ∈ En , then, ⋆. ci1 · · · cin ξ ⋆ ρ = δin n−1 · · · δi1 0 δρξ. s. µvnξ⋆ µv ξ ⋆. s(ξ ⋆ ).. 0. ⋆. Proof. Since ξ , ρ ∈ E then the only c operator that could give a non-zero result when applied the path ξ ⋆ ⋆ ρ is cn−1 (thus im = n − 1), indeed it gives a nonzero result only if given a certain elementary path ξI⋆ appearing in the expression of ξ ⋆ there is a corresponding elementary path ρI appearing in the expression of ρ such that the first step in ξI (i.e. the inverse of the last step of ξI⋆ ) coincides with the Rev. Un. Mat. Argentina, Vol 51-2.

(44) 168. R. TRINCHERO. ⋆. ⋆. ⋆. first step of ρI . More precisely, if ξI⋆ = (v0ξ , v1ξ , · · · , vnξ ) and ρI = (v0ρ , v1ρ , · · · , vnρ ), then s µvnξ⋆ ξ⋆ ξ⋆ ξ⋆ ⋆ cn−1 (ξI ⋆ ρI ) = δvnξ⋆ vρ δvξ⋆ vρ , v1ρ , · · · , vnρ ) (v0 , v1 , · · · , vn−1 0 n−1 1 µv ξ ⋆ n−1 s µvnξ⋆ ⋆ ξ ⋆ ρI,n−1 . = δvnξ⋆ vρ δvξ⋆ vρ 0 n−1 1 µvξ⋆ I,n−1 n−1. The first delta function appears because the concatenation ξ ⋆ ⋆ ρ should not vanish, the second from the definition of the c operator and the last equality is just a ⋆ definition of the path ξI,n−1 ⋆ ρI,n−1 . Next consider the application of a c-operator ⋆ to ξI,n−1 ⋆ ρI,n−1 , in a similar fashion, only cn−2 (thus im−1 = n − 2) gives a non-zero result, which is v u µ ξ⋆ ⋆ ⋆ u v ξ⋆ ⋆ cn−2 (ξI,n−1 ⋆ ρI,n−1 ) = δvξ⋆ vρ t n−1 (v0ξ , v1ξ , · · · , vn−2 , v2ρ , · · · , vnρ ) n−2 2 µv ξ ⋆ n−2 v u µ ξ⋆ u v ⋆ = δvξ⋆ vρ t n−1 ξI,n−2 ⋆ ρI,n−2 . n−2 2 µv ξ ⋆ n−2. Proceeding in this way and collecting the contribution of each elementary term finally leads to s µvnξ⋆ ⋆ ci1 · · · cim ξ ⋆ ρ = δim n−1 · · · δi1 0 δρξ s(ξ ⋆ ). µv ξ ⋆ 0.  Using the above result leads to Proposition 43. Definition (9.2) satisfies (9.1) with s µs(ω) µr(ξ) . F (ξ, ω) = µr(ω) µs(ξ) Proof. Replacing the ansatz (9.2) in the last axiom in (9.1) leads to X X F (ξ, ξc )(ξc⋆ ⊗ ξ ⋆ )·(ξc ⊗ ξd )⊠ ξd ⊗ ω = v ⊗ u ⊠ (ξ ⊗ ω)·(u ⊗ v ′). (9.5) v,u,v ′ ∈E0. ξc ,ξd ∈E. Employing the definition of the product and the fact that (ξ ⊗ ω) · (u ⊗ v ′ ) = δr(ξ)u δr(ω)v′ (ξ ⊗ ω) shows that (9.5) is equivalent to X X F (ξ, ξc )P (ξc⋆ ⋆ ξc ⊗ ξ ⋆ ⋆ ξd ) = δξd ξ v ⊗ r(ξ). (9.6) ξc ∈E. v∈E0. Choosing the factor F (ξ, ξc ) to be of the form s F (ξ, ξc ) = α(ξ). Rev. Un. Mat. Argentina, Vol 51-2. µv ξ c 0. µvnξc. ,.

(45) PATHS ON GRAPHS AND ASSOCIATED QUANTUM GROUPOIDS. 169. the l.h.s. of this last equation is given by X F (ξ, ξc )P (ξc⋆ ⋆ ξc ⊗ ξ ⋆ ⋆ ξd ) = ξc ∈E. X. = α(ξ). v(=r(ξc )). X. = α(ξ). v∈E0 ,ρ∈E. = α(ξ). (0) ⋆ v ⊗ ρ(ρ, c0 · · · cn−2 cn−1 (Π(0) n ⋆ Πn )ξ ⋆ ξd ). X. v ⊗ ρ(ρ, r(ξ))δξξd. v∈E0 ,ρ∈E. s. = α(ξ). † † (0) † ⋆ v ⊗ ρ((Π(0) n ⋆ Πn )cn−1 cn−2 · · · c0 ρ, ξ ⋆ ξd ). X. v∈E0 ,ρ∈E. = α(ξ). † † (0) † ⋆ P ((Π(0) n ⋆ Πn )cn−1 cn−2 · · · c0 (v) ⊗ ξ ⋆ ξd ). s. µs(ξ) µr(ξ). X µs(ξ) δξξd v ⊗ r(ξ), µr(ξ) v∈E0. where in the first equality we have employed (9.4) of proposition 41, the second equality involves the definition of the projector P , the hermiticity of the projector (0) (0) (Πn ⋆ Πn ) was employed in writing the third equality, the fact that ξ ⋆ and ξd are already essential and proposition 42 were employed in the fourth equality. Thus choosing s s µr(ξ) µr(ξ) µs(ξc ) ⇒ F (ξ, ξc ) = α(ξ) = µs(ξ) µs(ξ) µr(ξc ) leads to the result.  References [1] A. Cappelli, C. Itzykson and J.B.Zuber, Nucl. Phys. B 280 (1987) 445-465; Comm.Math. Phys. 113 (1987) 1-26. 147 [2] P. Di Francesco and J.-B. Zuber, in Recent developments in Conformal Field Theory, Trieste Conference, 1989, S. Randjbar-Daemi, E. Sezgin and J.-B. Zuber eds, World Sci, 1990. P. Di Francesco, Int. J. of Mod. Phys. A7, 92, 407. 147 [3] A. Ocneanu, The Classification of subgroups of quantum SU(N), Lectures at Bariloche Summer School, Argentina, Jan.2000, AMS Contemporary Mathematics 294, R. Coquereaux, A. García and R. Trinchero eds. 147 [4] E. Isasi and G. Schieber, J.Phys.A40(2007)6513-6538. 147 [5] G. Böhm and K. Szlachányi, A coassociative C ∗ -quantum group with non-integral dimensions, Lett. Math. Phys. 38 (1996), no. 4, 437–456, math.QA/9509008 (1995). 147 [6] A. Ocneanu, Paths on Coxeter diagrams: from Platonic solids and singularities to minimal models and subfactors. Notes taken by S. Goto, Fields Institute Monographs, AMS 1999, Rajarama Bhat et al, eds. 147 [7] R. Coquereaux, G. Schieber, Twisted partition functions for ADE boundary conformal field theories and Ocneanu algebras of quantum symmetries, J. of Geom. and Phys. 781 (2002), 1-43, hep-th/0107001. 147. Rev. Un. Mat. Argentina, Vol 51-2.

(46) 170. R. TRINCHERO. [8] R. Coquereaux and M. Huerta, Torus structure on graphs and twisted partition functions for minimal and affine models. J. of Geom. and Phys., Vol 48, No4, (2003) 580-634, hep-th/0301215. 147 [9] R. Trinchero, Adv. Theor. Math. Phys. vol. 10 , 1(2006). 147, 148 [10] V.B. Petkova and J.-B. Zuber, The many faces of Ocneanu cells, Nucl. Phys. B 603(2001) 449, hep-th/0101151. 147 [11] R. Coquereaux and R. Trinchero, On Quantum symmetries of ADE graphs, Adv. Theor. Math. Phys. vol. 8(2004)189-216. 147, 162 [12] R. Coquereaux, Racah - Wigner quantum 6j Symbols, Ocneanu Cells for ANdiagrams, and quantum groupoids, hep-th/0511293 - Journal of Geometry and Physics 57, 2 (2007) 147 [13] A. Ocneanu; Quantized groups, string algebras and Galois theory for algebras. Warwick (1987), in Operator algebras and applications, Lond Math Soc Lectures note Series 136, CUP (1988). 147 [14] D. Evans and Y. Kawahigashi, Quantum symmetries and operator algebras, Clarendon Press, 1998. 147 [15] Ph. Roche, Ocneanu cell calculus and integrable lattice models Commun.Math.Phys. 127, 395-424 (1990). 147 [16] J.-B. Zuber, CFT, ADE and all that Lectures at Bariloche Summer School, Argentina, Jan 2000. AMS Contemporary Mathematics 294, R. Coquereaux, A. García and R. Trinchero eds., hep-th/0006151. 150 [17] C. Itzykson and J. B. Zuber, Quantum Field Theory, McGraw-Hill, 1980. 153 [18] R. Coquereaux and A. Garcia, On bialgebras associated with paths and essential paths on ADE graphs, math-ph/0411048 - Intern. J. of Geom. Meth. in Mod. Phys., Vol 2, No 3, 2005, 1-26 158. R. Trinchero Centro Atómico Bariloche e Instituto Balseiro, Bariloche, Argentina. trincher@cab.cnea.gov.ar Recibido: 30 de marzo de 2010 Aceptado: 3 de mayo de 2010. Rev. Un. Mat. Argentina, Vol 51-2.

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On the Betti numbers of filiform Lie algebras over fields of characteristic two
For any nilpotent Lie algebra of Vergne type of dimension at least 5 over F, there exists a non-isomorphic Lie algebra of Vergne type having the same Betti numbers.. Note that
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12
Strongly transitive geometric spaces: Applications to hyperrings
Strongly transitive geometric spaces associated to hyperrings Freni [11] introduced the notion of strongly transitive geometric space on hypergroups and proved that the relation ∼= γ is
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11
On the notion of a ribbon quasi Hopf algebra
In this way, we arrive at the opposite and coopposite quasi-Hopf algebra Aop cop , which is again a quasi-Hopf algebra with respect to the following structure elements: Its counit and
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