dahmane hammaoui- the smallest ocneanu quantum groupoid of su(3) type

8/3/2019 Dahmane Hammaoui- The Smallest Ocneanu Quantum Groupoid of SU(3) Type

1/14

Dahmane Hammaoui

The Arabian Journal for Science and Engineering, Volume 33, Number 2CDecember 2008 225

Received 31 May 2008; Accepted 11 November 2008

THE SMALLEST OCNEANU QUANTUM

GROUPOID OF SU(3) TYPE

ABSTRACT

Ocneanu quantum groupods are a type of weak Hopf algebras which can be defined for every Lie group, andare at the core of the study of the quantum symmetries related to higher Coxeter graphs. In the present paperI analyze the whole theory for this particular type of weak Hopf algebras or Ocneanu double triangle algebra in a very simple case: the A1 Ocneanu quantum groupod of SU(3) type. This small example is interesting tostudy because it allows to explicitly describe the various structures of the theory.

MSC (2000): 81R50

Key words: Coxeter-Dynkin graphs, fusion and graph algebras, module graphs, modular invariants, quantumsymmetries, Ocneanu double triangle algebras

E-mail: [email protected] or [email protected]

:

Ocneanu Coxeter. Hopf Lie. Hopf

Ocneanu :OcneanuA1 SU(3). .

Dahmane Hammaoui

Equipe de Physique Mathematique et PlasmasLaboratoire de Physique Theorique

Physique des Particules et Modelisation (LPTPM)

Departement de Physique, Faculte des Sciences, Oujda, Morocco

and

Centre de Classes Preparatoires aux Grandes Ecoles dIngenieurs -CPR- Tanger, Morocco


2/14

Dahmane Hammaoui

The Arabian Journal for Science and Engineering, Volume 33, Number 2C December 2008 226

1. INTRODUCTION

Higher Coxeter graphs of SU(N) type were introduced by A. Ocneanu as a generalization to higher order of

the usual ADE Coxeter-Dynkin diagrams [1]. Their classification is an interesting problem both in physics andin mathematics and is complete only for SU(N) when N = 2, 3, and 4. In mathematics these graphs appear invarious fields such as representation theory of quantum groups, (weak) Hopf algebras, classification of semi-simpleLie algebras, operator algebras, nets of subfactors, category theory, and bimodules [2, 3, 4, 5, 6, 7, 8, 1]... Inphysics, many models are based on this kind of graph, such as lattice integrable models in statistical mechanicsor models of quantum gravity in string theory and D-branes [9, 10, 11, 12, 13, 14]. From a conformal fieldtheoretical viewpoint higher Coxeter graphs of the SU(N) type are related to the classification of modularinvariants of conformal models [15, 16, 17, 18] and enable us to construct twisted partition functions [19, 20, 21].In the topological field theoretical framework, these graphs provide the necessary data to compute the set of 3jand 6j-symbols in many theories of strings [22].

Quantum geometry of ADE and generalized Coxeter-Dynkin graphs (introduced by A. Ocneanu [8]) describesquantum symmetries of these graphs together with their corresponding Ocneanu graphs, which index the defectlines of the CFT and brings out new structures of weak Hopf algebras called algebras of double triangles. Analgebraic method was adopted by R. Coquereaux [23] to build the Ocneanu algebra of quantum symmetries of agiven graph G of SU(N) type. Following this procedure, Ocneanu graphs and algebras of quantum symmetrieswere either recovered (when they were already known) or established for all members of SU(2) type [19, 24, 25],of SU(3) type [26, 27, 28] and exceptional cases of SU(4) system [29, 30]. In this way to every pair of graphs ofan SU(N) system with same generalized Coxeter number (namely a graph G and the corresponding graph A(G)which describes its fusion algebra), one associates an algebra of quantum symmetries. Its elements are calledirreducible quantum symmetries that one can add and multiply like the irreducible representations of groupsand the Ocneanu graph encodes this algebraic structure. One can also start from a given modular invariantand applying the modular splitting formula [8, 25, 28] to recover this algebra of quantum symmetries. Thefusion algebra and the Ocneanu algebra of quantum symmetries describe the theory of representations of the so-called double triangle algebra (DTA). This later is a bialgebra (more precisely a weak Hopf algebra or Ocneanuquantum groupod) which is related to the geometry of paths on the graph G. The paths are of two type: the

horizontal paths indexed by the irreducible representation (irreps) of the fusion algebra and the vertical onesindexed by the irreducible quantum symmetries. This property of being a weak Hopf algebra characterizingthis Ocneanu bialgebra was given in [31] in a framework of quantum cohomology and was investigated in [32]by considering solutions to the big pentagon equation. To ensure some consistency conditions for the productsand coproducts and the pairing between dual spaces one needs to introduce fundamental objects called Ocneanucells (a kind of generalized 6j-symbols). Because of the hard and technical task to compute these Ocneanu cells,the full construction and study of the Ocneanu bialgebras are established only for the Ak graphs belonging tothe ADE members of SU(2) system [33, 34, 35]. However, there is much information on quantum groupodsrelated to the Di Francesco-Zuber system of graphs associated with the SU(3) system in [26, 27]. Note that anice relation between these construction and face models in statistical mechanics can be found in [36].

The purpose of this paper is to give a brief description of the Ocneanu bialgebras where we focus more on thealgebra and coalgebra structures which are the special topics of this number of AJSE. The concrete results arerelative to the simplest (but not trivial) case of SU(3) system: the A1 graph. Note that we need no knowledge

or use of Ocneanu cells since their values are all equal to 1 in this particular case. The article is organized asfollows: the first part deals with the necessary background on graphs and their quantum symmetries where aredefined the fusion algebra A (G) of a given graph G, the associated Ocneanu algebra of quantum symmetriesOc (G), and the properties of G as an A-module and as an Oc (G)-module and the connection with the A (G)and Oc (G) algebras on one side and the DTA structure on the other side. The second part is devoted to the

Ocneanu bialgebras (there are two dual bialgebras denoted B and B which are isomorphic) where are definedhorizontal and vertical double triangles and the associated algebras and coalgebras structures. As an explicitapplication we give a full construction of the DTA related to the A1 case.

THE SMALLEST OCNEANU QUANTUM GROUPOID OF SU(3) TYPE


3/14

Dahmane Hammaoui


2. BACKGROUND ON GRAPHS AND RELATED ALGEBRAIC STRUCTURES

Given a modular invariant M i.e. a matrix associated to a modular invariant partition function Z in RCFT(see [15, 16] for the SU(2) series or [17, 18] for the SU(3) series but for higher SU(N) series there are no completeclassification), one can associate at least one higher Coxeter graph G of SU(N) type [1]. The graph G is amodule graph on the fusion algebra A (G) related to the graph Ak with same generalized Coxeter number as

G. The Ak graph has dA vertices denoted ,,..., while for the G graph there are dG vertices that we denotesby a, b,c... The structure of A-module is encoded in non-negative integers F ba which constitute a non-negativeinteger valued-matrix representation (nim-rep) of dimension dG dG for the fusion algebra [23, 19, 18]. Theinteger F ba is just the number of horizontal essential paths on G, associated to the irreducible representation ofAk, starting from the vertex a and arriving at the vertex b. The graph G is also a module graph on its algebraOc(G) of quantum symmetries related to the so-called Ocneanu graph that we denote by (G). Vertices of the (G) graph are denoted x,y,z.... The structure ofOc (G)-module is encoded in coefficients S bxa which constitutea nim-rep of dimension d(G) d(G) for the Ocneanu algebra of quantum symmetries [23, 19, 24, 26, 27, 30].

The integer S bxa is the number of vertical paths on G, associated to the irreducible quantum symmetry x ofOc(G), starting from the vertex a and arriving at b.

2.1. The Fusion Algebra and the Graph of A Type

Ak graphs are the Weyl alcoves of SU(N) type truncated at some level k and are characterized by the(dual) Coxeter number = N+ k. The vertices , , . . . denote irreducible representations (irreps) of quantumsub-groups SU(N)k, at a root of unity q = ei/. The Ak graphs encode the tensor product of irreps inheritedfrom fusion of primary fields in CFT [37]:

Vi Vj = kNk

ij Vk,

where V are irreducible representations of the chiral algebra, an extension of the Virasoro algebra, of the RCFT.

For instance, the Weyl alcoves of SU(3) case are:

Ak = { = (1, 2) = 11 + 22 | 0 1 + 2 k, 1, 2 N} .

where 1 and 2 are the fundamental weights of the SU(3) Lie group and 1, 2 are the corresponding Dynkin

labels. (0, 0) is the unit representation which index the unit vertex of Ak and is related to the vacuum state,(1, 0) is the fundamental generator (irrep) of SU(3)k and (0, 1) is its conjugate. The cardinality of Ak isdAk = (k + 1)(k + 2)/2.

The nim-reps N kij are given by the Verlinde formula [38]:

N kij =

mAk

SimSjmSkm

S0m.

where S are the entries of the symmetric, unitary Smatrix of the modular transformations of the charac-ters i() = T r e

2i(L0c/24) such that (1/) =

AkS(). The other generator T of the modular

group is defined by its action on the characters by ( + 1) =

AkT(). A dAk dAk -dimensional

matrix representation of the fusion algebra (also called the Verlinde algebra) is given by the following formula:

NN = AkNN .

Each matrix N is associated to a vertex = (1, 2) of the graph Ak. To determine the N, it is convenientto use the known recurrence relation for coupling of irreducible SU(3) representations: N(0,0) = IdAk , N(1,0) =A(Ak) is the adjacency matrix of the graph Ak (the adjacency matrix A is reading directly from the graph,conversely the knowledge of the A enable one to represent the graph itself), and

N(,) = N(1,0) N(1,) N(1,1) N(2,+1) if = 0

N(,0) = N(1,0) N(1,0) N(2,1)

N(0,) = (N(,0))tr


4/14

Dahmane Hammaoui


2.2. Quantum Symmetries and Ocneanu Graphs

As we shall see in the second part, the Ocneanu bialgebra under study is a semi-simple graded algebra ofendomorphisms of essential paths. It has two diagonal block decompositions in simple blocks. The first typeof blocks are labeled by Young diagrams (irreps) of the graph Ak and the fusion algebra Ak is interpretedas the algebra spanned by minimal central projections related to these blocks. While the second type of blocks

are indexed by the irreducible quantum symmetries x of the Ocneanu graph (G) and the related algebra ofquantum symmetries Oc (G) is the algebra spanned by minimal central projections associated to these secondblocks. Another interpretation of quantum symmetries in the framework of twisted chiral conformal field theoryis as follows: The vertices x, y, z . . . of the Ocneanu graph encode defect lines of the theory, and their fusion isdescribed by the Ocneanu algebra of quantum symmetries.

A technical method to determine the algebra of quantum symmetries of a graph is the modular splittingformula based only on the knowledge of M(G) and the fusion matrices N of A(G):

N

N

M =

z(W0z)(Wz0) .

which enables us to compute all toric matrices1 (of size dA dA) W0z or Wz0 with one defect line z. itsgeneralization gives a relation between twisted toric matrices with two defect lines.

z(Wxz)(Wzy ) = N N (Wxy) .The others Wxy, which give the generalized twisted partition functions of the theory with two defect lines (sayx and y), are given by

z(Wxz)Wz0 = NWx0(N)tr.

A faithful anti-representation of the Ocneanu algebra Oc(G) is carried by N-valued matrices called Ocneanumatrices Ox, of size d d, indexed by the vertices x of (G):

OxOy =

z Oz

yxOz.

To determine the adjacency matrices of the Ocneanu graph (G) related to the fundamental generators, itis convenient to introduce the following matrices V of size d d such that their entries satisfy to (V)xy =(Wxy). These are the so-called double annular matrices and constitute a nim-rep for the double fusion algebraAk Ak

VV =

N

N

V

What is special in these notions of double fusion algebra is that the fundamental generators of the Ocneanugraph are related to double annular matrices: Vf0 and V0f where f are the fundamental irreps of the SU(N)Lie group. For the SU (3) case, where f is either (1, 0) or its conjugate (0, 1), the Ocneanu graph (G) hasfour chiral fundamental generators2 associated to the following adjacency matrices:

V(1,0)(0,0) = O1L V(0,0)(1,0) = O1RV(0,1)(0,0) = O1L V(0,0)(0,1) = O1R

and the unit vertex of (G) is associated to the identity matrix V(0,0)(0,0) = IdA.

2.2.1. Module Structures Related to the Graph G

On the one hand, the graph G is a module over the fusion algebras A(G)

A G G.a

b F

bab

1The toric matrix associated to the unit vertex is the modular invariant itself: W00 = M.2When the graph G is a graph ofA type, one has only one fundamental generator and its conjugate. Furthermore the Ocneanu

graph (A) is the graph A itself and all types of the cited sets of matrices coincide with the set of fusion matrices N.


5/14

Dahmane Hammaoui


This action ofA-module on vertices ofG is encoded in the nim-reps (F)ab which provide solutions to the Cardyequation in boundary CFT [39], and where the boundary states of the theory are represented by the verticesa , b , c . . . of G. The matrices F are called the annular matrices such that F(0,0) = IdG , F(1,0) = A(G) i.e. theadjacency matrix of the graph G satisfies the same recurrence formula as the fusion matrices N

FF =

N

F .

On the other hand, the graph G is also a module over the Ocneanu algebra of quantum symmetries Oc(G).This structure is encoded in the set of nim-reps (Sx)ab :

Oc(G) G Gx.a

b S

bxab

The set of matrices Sx, called dual annular matrices, forms a new anti-representation of the Ocneanu algebrasatisfying to

SxSy =

z Oz

yxSz.

2.3. A Weak Hopf Algebra Structure: The Double Triangle Algebra

There are two type of paths on a given graph G, the essential horizontal paths indexed by Young frames of

A(G) and the vertical ones labeled by Ocneanu generators of Oc(G). Endomorphisms of paths define two dualgraded algebras denoted B(G) and B(G) and represented by horizontal and vertical double triangles respectively(see [8] and [32]). These are both finite dimensional algebras with isomorphic underlying vector spaces and eachone of the two algebra structures can be traded against a coalgebra structure on its dual. Equipped with amultiplication law, a unit, a coproduct, a counit, and an antipode, the bialgebra B(G) (and of course B(G))satisfies all the axioms of weak Hopf algebras cited in [2, 3] for instance. These are a particular kind of weakHopf algebras called Ocneanu quantum groupods. The representation theory of B(G) is encoded by the fusion

algebra A(G) and the representation theory of B(G) is realized by the Ocneanu algebra of quantum symmetriesOc(G). Many explicit examples are known only for the SU(2) case as in [33, 34, 35, 36], in the following we tryto treat in a full and explicit way these structures for the first member of the SU(3) system.

3. THE OCNEANU BIALGEBRA STRUCTURES

In what follows, we give explicit results on the simplest (but not trivial) case of SU(3) system: the A1 graph3.The unit vertex is 1 (0, 0), the left generator is 3 (1, 0) and the right one is 3 (0, 1). The level of thegraph is k = 1 and its dual Coxeter number is = k + 3 = 4. The following figure (Figure 3) shows the graphA1 and its adjacency matrix A (A1).

ddd

1 3

3

E

st re

re

A (A1) =

0 1 00 0 1

1 0 0

Figure 1: The A1 graph and its adjacency matrix

The fusion matrices of the Verlinde algebra A1 are:

N1 = I3 =

1 . .. 1 .

. . 1

, N3 = A (A1) =

. 1 .. . 1

1 . .

, N3 = A (A1)tr =

. . 11 . .

. 1 .

.

For the Ak cases, the graph Ak and the Ocneanu graph (Ak) coincide, and more especially all the followingmatrices are the same: Ni = Fi = Oi = Si .

3For brevity, we denote the generators by their classical dimensions i.e. 1, 3, and 3.


6/14

Dahmane Hammaoui


3.1. Essential Paths on G and the Double Triangle Algebras

3.1.1. General Data

Essential horizontal paths (or horizontal triangle) ab of type going from a to b span a vector space Hpaths(G)graded by : Hpaths(G) = AH with cardinality

Ad where d =

a,bG(F)ab.

Vertical paths (or vertical triangles) abx of type x going from a to b span a vector space Vpaths(G) gradedby x: Vpaths(G) = xVx. Its dimension is

x dx where dx =

a,bG (Sx)ab.

ab =

dda b

= dda b

T

xab =

dda

b

x=

dd

a

b

xE

Figure 2:

Now we consider the vector spaces of endomorphisms on essential paths:

B = B =

H(G)

H(G)

B = x

Bx =

x Vx(G)

Vx(G).

Equipped with a product called vertical product defined as a composition of endomorphism of essentialpaths, B is endowed with an algebra structure called algebra of double horizontal triangles. In the same way,B is the algebra of double vertical triangles equipped by a product called horizontal product. Moreover theproduct in B allows us to define a coproduct for its dual B, and vice-versa, so that we finally obtain twodually paired bialgebras, which are both semi-simple and co semi-simple.

A basis for the algebra (B, ) is defined by {e } where e stands for ab cd and are represented by matrixunits. For the dual algebra

B, a basis is defined by f where f means abx cdx . These are the algebrasof double triangles or by duality the algebras of diffusion graphs.

e =dd

dd

a a

b b

T

s

dd

dd

a a

b b

fx =

dd

dd

x

c

c

d

d

dd

dd

x

c

c

d

d

'

s

Figure 3:

3.1.2. The A1 Case

In the A1 case, the oriented horizontal essential paths lpq, assigned to Young frame l, starting from a vertexp and ending at a vertex q are denoted as pql or pl, and span the vector space Hl. The graded vector space ofall horizontal essential paths is the direct sum: H = lA1Hl.

H1 = 111,

133,

133 =

111,

331,

331 = 11, 31, 31H3 = 313, 333, 331 = 133, 333, 313 = 13, 33, 33

H3 =

313

, 331, 333

=

133,

313,

333

=

13, 33, 33

The algebra of horizontal double triangles or by duality the algebra of vertical diffusion graphs is:

B = lA1Bl = lA1Hl Hl = (H1 H1) (H3 H3) (H3 H3)The elements el for each Bl are shorten written as e

lpq = pl ql.


7/14

Dahmane Hammaoui


B1 =

e111, e113, e

113

, e131, e133, e

133

, e131

, e133

, e133

B3 =

e311, e

313, e

313

, e331, e333, e

333

, e331

, e333

, e333

B3 =

e311, e

313, e

313

, e331, e333, e

333

, e331

, e333

, e333

dd

dd

1

1 1

1 1

Ts

dd

dd

1

1 1

3 3

Ts

dd

dd

1

1 1

3 3

Ts

dd

dd

1

3 3

1 1

Ts

dd

dd

1

3 3

3 3

Ts

dd

dd

1

3 3

3 3

Ts

dd

dd

1

3 3

1 1

T

s

dd

dd

1

3 3

3 3

T

s

dd

dd

1

3 3

3 3

T

s

B =

dd

dd

3

1 3

1 3

Ts

dd

dd

3

1 3

3 3

Ts

dd

dd

3

1 3

3 1

Ts

dd

dd

3

3 3

1 3

Ts

dd

dd

3

3 3

3 3

Ts

dd

dd

3

3 3

3 1

Ts

dd

dd

3

3 1

1 3

T

s

dd

dd

3

3 1

3 3

T

s

dd

dd

3

3 1

3 1

T

s

dd

dd3

1 3

1 3

Ts

dd

dd3

1 3

3 1

Ts

dd

dd3

1 3

3 3

Ts

dd

dd3

3 1

1 3

Ts

dd

dd3

3 1

3 1

Ts

dd

dd3

3 1

3 3

Ts

dd

dd

3

3 3

1 3

T

s

dd

dd

3

3 3

3 1

T

s

dd

dd

3

3 3

3 3

T

s

Figure 4:

The dimension of each block Bl of B is dim Bl = d2l =

m,n (Fl)mn

2and dim B =

l d

2l = 27. A basis

for the algebra B is made of matrix units (pql) of dimension 9 9 and take the value4 1 at the indicated place

and 0 elsewhere:

elpq

=

e111 e113 e113e131 e133 e133

e131

e133

e133

e311 e313 e313e331 e333 e333

e331

e333

e333

e311 e313 e

313

e331 e333 e

333

e331

e333

e333

The oriented vertical essential paths pqL , associated to an Ocneanu vertex x, represented here by L, startingfrom a vertex p and ending at a vertex q are denoted pq

L or pL, and span the vector space VL. The global

graded vector space of all vertical essential paths is the direct sum: V = L(A1)VL.

V1 =

111 , 331 ,

331

=

111, 331, 331

=

11,31,31V3 =

113 ,

333 ,

333

=

133, 333, 333

=

13,33,33V3 =

11

3, 33

3, 33

3

=

133, 313, 333

=

13,33,33The algebra of vertical double triangles or by duality the algebra of horizontal diffusion graphs

B is the dual

vector space of the vector space B, it is a graded algebra which is isomorphic5 to the algebra B such that:

B = B1 B3 B3.

4For example the matrix unit (313) is a square matrix of size 9 9, where the unique non-zero value is 1 at the 5 th line and 4th

column.5In fact B and B describe the same algebra structure.


8/14

Dahmane Hammaoui


An appropriate basis of B is made of vertical double triangles represented by matrix units fpqL = pL qL

(fpqL ) =

f

111 f

131 f

131

f311 f331 f

331

f311 f331 f

331

f

113 f

133 f

133

f313 f333 f

333

f313 f333 f

333

f

113

f133

f133

f313

f333

f333

f313

f333

f333

3.2. A Bialgebra Structure

3.2.1. The Algebra (B, )

The explicit definition of the vertical products on B is written as: e e

= e . In the A1 casewe obtain the following results

e111 e

113 e

113

e131 e133 e

133

e131

e133

e133

e111 e111 e

113 e

113

. . . . . .

e113 . . . e111 e

113 e

113

. . .

e113

. . . . . . e111 e113 e

113

e131 e131 e

133 e

133

. . . . . .

e133

. . . e131

e133

e133

. . .

e133

. . . . . . e131 e133 e

133

e131

e131

e133

e133

. . . . . .

e133

. . . e131

e133

e133

. . .

e133

. . . . . . e131

e133

e133

e311 e

313 e

313

e331 e333 e

333

e331

e333

e333

e311 e311 e

313 e

313

. . . . . .

e313 . . . e311 e

313 e

313

. . .

e313

. . . . . . e311 e313 e

313

e331 e331 e

333 e

333

. . . . . .

e333 . . . e331 e

333 e

333

. . .

e

3

33. . . . . .

e

3

31 e

3

33 e

3

33e3

31e3

31e3

33e3

33. . . . . .

e333

. . . e331

e333

e333

. . .

e333

. . . . . . e331

e333

e333

dd

dd1

1

1

1

1

'

s

dddd1

1

1

3

3

'

s

dddd1

1

1

3

3

'

s

dd

dd1

3

3

1

1

'

s

dddd1

3

3

3

3

'

s

dddd1

3

3

3

3

'

s

dd dd1

3

3

1

1

's dd dd1

3

3

3

3

's dd dd1

3

3

3

3

's

B =dd

dd3

3

1

3

1

'

s

dddd3

3

1

3

3

'

s

dddd3

3

1

1

3

'

s

dd

dd3

3

3

3

1

'

s

dddd3

3

3

3

3

'

s

dddd3

3

3

1

3

'

s

dd dd3

1

3

3

1

's dd dd3

1

3

3

3

's dd dd3

1

3

1

3

's

dd

dd3

3

1

3

1

'

s

dddd3

3

1

1

3

'

s

dddd3

3

1

3

3

'

s

dd

dd3

1

3

3

1

'

s

dddd3

1

3

1

3

'

s

dddd3

1

3

3

3

'

s

dd dd3

3

3

3

1

's dd dd3

3

3

1

3

's dd dd3

3

3

3

3

's

Figure 5:


9/14

Dahmane Hammaoui


e311 e

313 e

313

e331 e333 e

333

e331

e333

e333

e311 e311 e

313 e

313

. . . . . .

e313 . . . e311 e

313 e

313

. . .

e313

. . . . . . e311 e313 e

313

e331 e331 e

333 e

3

33

. . . . . .

e333 . . . e331 e

333 e

333

. . .

e333

. . . . . . e331 e333 e

333

e331

e331

e333

e333

. . . . . .

e333

. . . e331

e333

e333

. . .

e333

. . . . . . e331

e333

e333

The unit element of (B, ) is

I = e111 + e133 + e

133

+ e311 + e333 + e

333

+ e311 + e333 + e

333

,

which can be written as I =

ll, l = 1, 3 and 3, where the i are the minimal central projections of B on Bi:

1 = e111 + e133 + e

133

3 = e311 + e333 + e

333

, 3 = e311 + e

333 + e

333

such that 2k

= k

, = , and k

Bk

= k

B = Bk

.

3.2.2. The Coalgebra (B,)

The existence of a product in B allows to define a coproduct : B B B in such a way to fulfill thecompatibility condition6 of a bialgebra structure between and

f f , e

=

ff , e .For the A1 case of SU(3), a coproduct7 is defined in such a way that it is a morphism for the product i.e.

(pql rsk) = (pql) (rsk). This procedure follows the definition involved in [34] for the A2 case of SU(2)type. From this obtained coproduct, we should deduce the definition of on the dual vector space. Explicitlywe obtain the following results

(e111) = e111 e

111 + e

311 e

333 + e

311 e

333

(e113) = e113 e

113 + e

313 e

333

+ e313 e331

(e113

) = e113

e113

+ e313

e331 + e313

e333

(e131) = e131 e

131 + e

331 e

333

+ e331 e313

(e133) = e133 e

133 + e

333 e

333

+ e333 e311

(e133

) = e133

e133 + e333

e331

+ e333

e313(e1

31) = e1

31 e1

31+ e3

31 e313 + e

331

e333

(e133

) = e133

e133

+ e333

e313

+ e333

e331

(e133) = e133 e133 + e333 e311 + e333 e311

(e311) = e111 e

311 + e

311 e

133 + e

311 e

333

(e313) = e113 e

313 + e

313 e

133

+ e313 e331

(e313

) = e113

e313

+ e313

e131 + e313

e333

(e331) = e131 e

331 + e

331 e

133

+ e331 e313

(e333) = e133 e

333 + e

333 e

133

+ e333 e311

(e333

) = e133

e333 + e333

e131

+ e333

e313(e3

31) = e1

31 e3

31+ e3

31 e113 + e

331

e333

(e333

) = e133

e333

+ e333

e113

+ e333

e331

(e333) = e133 e333 + e333 e111 + e333 e311

6In general cases the values of cells appear in the pairing between both types of matrix units (respectively associated with theproducts and )

7The definition of this first coproduct was given by E. Isasi [40].


10/14

Dahmane Hammaoui


(e311) = e111 e

311 + e

311 e

333 + e

311 e

133

(e3131) = e113 e

313 + e

313 e

333

+ e313 e131

(e313

) = e113

e313

+ e313

e331 + e313

e133

(e331) = e131 e

331 + e

331 e

333

+ e331 e113

(e333) = e133 e

333 + e

333 e

333

+ e333 e111

(e333) = e133 e333 + e333 e331 + e333 e113(e3

31) = e1

31 e3

31+ e3

31 e313 + e

331

e133

(e333

) = e133

e333

+ e333

e313

+ e333

e131(e3

33) = e1

33 e3

33+ e3

33 e311 + e

333

e111

Remark that the coproduct of the unit element of B is not I I, but of the form (I) =I(1) I(2) as

(I) = (e111 + e333

+ e333) (e111 + e

311 + e

311) + (e

133 + e

311 + e

333

) (e133 + e333 + e

333)

+ (e133

+ e333 + e311) (e

133

+ e333

+ e333

)

The counit of B can be defined as a C-valued linear map on B satisfying (e e) =

eI(1)

I(2)e

. For theA1 case we have

(e

l

pq) = 1, if l = 1

0, if l = 3, 3Note that the map is not an algebra homomorphism.

One can also define an antipode S on B as an algebra anti-homomorphism like a conjugation of elements

of B in the following way: S

e

= ke

, where =

ab = ba and k =

(a)(d)(b)(c) is a function of quantum

dimensions of the vertices of G. In the A1 case, since k = 1 (all the Perron-Frobenius components of the A1graph are equal to 1) we get S

elpq

= elqp.

3.3. The AlgebraB,

The explicit definition of the horizontal product

on

B is written as: fx

fx = xx f

. For the A1case it will involve in B the dual basis (elpq) of (elpq) instead of the basis of matrix units of the product: (f

pq

L).

Before manipulating with this change of basis in B we just mention the form of the unit element in terms of fpqL :I = f111 + f331 + f331 + f113 + f333 + f333 + f113 + f333 + f333Remark thatI = L L, L = 1, 3 and 3, where L are the minimal central projections of B, i.e. L = L = 2Land L(B) = BL.

To define the product of B for the A1 case, we use the compatibility condition fulfilled by the product ofB and the coproduct on B in terms of the dual basis (eab) and (eab):eab eab, (eab) = eabeab, eab

So, from the known explicit action of on elements elpq of B (A1), we deduce the non-zero values of

e111 e113 e113 e131 e133 e133 e131 e133 e133e111 e111 . . . . . . . .e113 . e113 . . . . . . .e113

. . e113

. . . . . .e131 . . . e131 . . . . .e133 . . . . e133 . . . .e133

. . . . . e133

. . .e131

. . . . . . e131

. .e133

. . . . . . . e133

.e133

. . . . . . . . e133


11/14

Dahmane Hammaoui


e311 e313 e313 e331 e333 e333 e331 e333 e333e311 . . . . e311 . . . .e313 . . . . . e333 . . .

e3

13. . .

e3

13. . . . .

e331 . . . . . . .

e331 .e333 . . . . . . . . e333e333

. . . . . . e333

. .e331

. e331

. . . . . . .e333

. . e333

. . . . . .e333

e333

. . . . . . . .

e311 e313 e313 e331 e333 e333 e331 e333 e333e311 . . . . . . . . e311e313 . . . . . . e313 . .e313

. . . . . . . e313

.

e331 . .

e331 . . . . . .

e333 e333 . . . . . . . .e333

. e333

. . . . . . .e331

. . . . . e331

. . .e333

. . . e333

. . . . .e333

. . . . e333

. . . .

3.4. The CoalgebraB,

In an analogous way a coproduct is defined on B by the use of the product in its dual B: f , e e = f, e e.For the present example, an explicit action of on the elements of the dual basis (elpq), such that thehomomorphism algebra (elpqelpq) = (elpq)(elpq) is fulfilled, reads as

(e111) = e111 e111(e113) = e113 e113(e113

) = e113

e113(e131) = e131 e131(e133) = e133 e133(e1

33) = e1

33e133(e1

31) = e1

31e1

31(e133

) = e133

e133

(

e1

33) =

e1

33

e1

33

(e311) = e311 e333(e313) = e313 e333(e313

) = e313

e331(e331) = e331 e333(e333) = e333 e333(e333

) = e333

e331(e3

31) = e3

31e313(e3

33) = e3

33e3

13

(

e3

33) =

e3

33

e311

(e311) = e311 e333(e313) = e313 e333(e313

) = e313

e331(e331) = e331 e333(e333) = e333 e333(e333

) = e333

e331(e3

31) = e3

31e313(

e3

33) =

e3

33

e3

13

(

e3

33) =

e3

33

e311

A counit of B can be defined as a C-valued linear map on B satisfying (f f) = fI(1)I(2)f, whereI(1) and I(2) come from the right hand side of acting on I(1) as: (I = I(1) I(2). For the A1 case, canbe defined on the dual basis as (elpq) = 1 if l = 10 if l = 3, 3 .Note that the map is not an algebra homomorphism.

One can also define an antipode S on B as an algebra anti-homomorphism by S(fx ) = kfx . For the A1case, we have simply Selpq = elqp.


12/14

Dahmane Hammaoui


4. CONCLUSION

The spaces B and B are both semi-simple and co semi-simple bialgebras which are isomorphic: B (, I, , )and B ,I, ,. Moreover, equipped with the antipodes S and S, B and B are weak Hopf algebras [2, 3] anddescribe the same algebra of double triangles [8]. In more general cases, for instance Ap, p > 1, in order toensure the axioms of weak Hopf algebras, one might to introduce the values of cells generalizing the quantum 3j

and 6j-symbols. One type of these cells are called Ocneanu cells and gives the pairing e , fx between thenot dual sets of basis

e

in B and

fx

in B, the other type ensure the associativity and the actions of

and [35, 32]. There are many technical difficulties in calculation of the set of cells for higher level; the studiedexample A1 is the only case of SU(3) for which the quantum groupod structure is explicitly determined. Agood amount of information about all graphs of SU(3) type is available in [26, 27] and several examples ofADEtype are studied for instance in [34, 33, 41, 35].

ACKNOWLEDGMENTS

This work emanated from a question asked by Dr E. Isasi, I would like to thank him together with ProfessorsR. Coquereaux, E. H. Tahri and G. Schieber for many useful and enjoyable discussions. I would like also to

thank Professor J. Abuhlail for his kind invitation to me to contribute to the AJSE.

REFERENCES

[1] A. Ocneanu, The Classification of Subgroups of Quantum SU(N). Lectures at Bariloche summer school,Quantum Symmetries in Theoretical Physics and Mathematics Argentina, January 2000, R. Coquereaux,AMS Contemp. 294., A. Garca and R. Trinchero (eds.).

[2] G. Bohm and K. Szlachanyi, A Coassociative C-Quantum Group with Non-Integral Dimensions, Lett.Math. Phys. 200 (1996), pp. 437456.

[3] F. Nill, Axioms for Weak Bialgebras, Diff. Geom. Quan. Phy. 334 (1998), pp. 148.

[4] J. Bockenhauer and D. E. Evans, Modular Invariants, Graphs and -Induction for Nets of Subfactors I,Comm. Math. Phys. 197 (1998), pp. 361386; II, Comm. Math. Phys. 200 (1999), pp. 57103; III, Comm.Math. Phys. 205 (1999), pp. 183228.

[5] J. Bockenhauer, D. E. Evans, and Y. Kawahigashi, On -Induction, Chiral Generators and ModularInvariants for Subfactors, Comm. Math. Phys. 208 (1999), pp. 429487.

[6] R. Longo and K. -H. Rehren, Nets of Subfactors, Rev. Math. Phys. 7 (1995), pp. 567497.

[7] V. Ostrik, Module Categories, Weak Hopf Algebras and Modular Invariants, Transform. Groups. 8 (2003),pp. 177206.

[8] A. Ocneanu, Paths on Coxeter Diagrams: From Platonic Solids and Singularities to Minimal Models andSubfactors, (Notes by S. Goto) In: Lectures on operator theory. Fields Institute Monographs 13, R. Bhat,

G. A. Elliott, and P. A. Fillmore (eds.). AMS Providence, 1999.[9] V. Pasquier, Ethiology of IRF Models, Comm. Math. Phys. 118 (1988), pp. 335364.

[10] V. Pasquier, Two-Dimensional Critical Systems Labelled by Dynkin Diagrams, Nucl. Phys. B. 285 [FS19](1987), pp. 162172.

[11] Ph. Roche, Ocneanu Cell Calculus and Integrable Lattice Models, Comm. Math. Phys. 127 (1990), pp.395424.

[12] P. Di Francesco and J.-B. Zuber, SU(N) Lattice Integrable Models Associated with Graphs, Nucl. Phys.B 338 (1990), pp. 602646.


13/14

Dahmane Hammaoui


[13] P. A. Pearce and Y. K. Zhou, Intertwiners and A-D-E Lattice Models, Int. J. Mod. Phys. B 7 (1993),pp. 36493705.

[14] N. Sochen, Integrable Models through Representations of the Hecke Algebras, Nucl. Phys. B 360 (1991),pp. 613640.

[15] A. Cappelli, C. Itzykson, and J. -B. Zuber, Modular Invariant Partition Functions in Two Dimensions,Nucl. Phys. B 280 (1987), pp. 445465.

[16] A. Cappelli, C. Itzykson, and J. -B. Zuber, The ADE Classification of Minimal and A(1)1 Conformal

Invariants Theories, Comm. Math. Phys. 113 (1987), pp. 120.

[17] T. Gannon, The Classification of Affine SU(3) Modular Invariants, Comm. Math. Phys. 161 (1994), pp.233263.

[18] D. Hammaoui, G. Schieber, and E. H. Tahri, Higher Coxeter Graphs Associated to Affine su(3) ModularInvariants, J. Phys. A: Math. Gen. 38 (2005), pp. 82598268.

[19] R. Coquereaux and G. Schieber, Twisted Partition Functions for ADE Boundary Conformal Field Theoriesand Ocneanu Algebras of Quantum Symmetries, J. Geom. Phys. 781 (2002), pp. 143.

[20] R. Coquereaux and M. Huerta, Torus Structure on Graphs and Twisted Partition Functions for Minimaland Affine Models, J. Geom. Phys. 48 (2003), pp. 580634.

[21] V. B. Petkova and J. -B. Zuber, Generalized Twisted Partition Functions, Phys. Lett. B 504 (2001), pp.157164.

[22] J. Fuchs, I. Runkel, and C. Schweigret, TFT Construction of RCFT Correlators I: Partition Functions,Nucl. Phys. B 646 (2002), pp. 353497; II: Unoriented World Sheets, Nucl. Phys. B 678 (2004), pp.511637; III, Simple Currents, Nucl. Phys. B 694 (2004), pp. 277353; IV: Structure Constants andCorrelations Functions, Nucl. Phys. B 715 (2005), pp. 539638.

[23] R. Coquereaux, Notes on the Quantum Tetrahedron, Moscow Math. J. 2. (2002), pp. 4180; Notes onthe Classical Tetrahedron: a Fusion Graph Algebra Point of View, http://www.cpt.univ-mrs.fr/coque/.

[24] R. Coquereaux and G. Schieber, Determination of Quantum Symmetries for Higher ADE Systems fromthe Modular T Matrix, J. Math. Phys. 44 (2003), pp. 38093837.

[25] R. Coquereaux and E. Isasi, On Quantum Symmetries of the Non-ADE Graph F4, Adv. Theor. Math.Phys. 8 (2004), pp. 955985.

[26] R. Coquereaux, D. Hammaoui, G. Schieber and E. H. Tahri, Comments About Quantum Symmetries ofSU(3) Graphs, J. Geom. Phys. 57 (2006), pp. 269292.

[27] D. Hammaoui Geometrie quantique dOcneanu des graphes de Di Francesco-Zuber associes aux modeles

conformes de type su(3), These de Doctorat national, LPTPM, Oujda University, Morocco, January 2007.

[28] E. Isasi and G. Schieber, From Modular Invariants to Graphs: The Modular Splitting Method, J. Phys.A: Math. Theor 40 (2007), pp. 6513-6537.

[29] R. Coquereaux and G. Schieber, From Conformal Embeddings to Quantum Symmetries: An Excep-tional SU(4) Example, IOP, Journal of Physics, Conference Series: International Conference on Non-Commutative Geometry and Physics, 23-27 April 2007, LPT, Universite Paris XI, France. Arxiv: math-ph/07101397.

[30] R. Coquereaux and G. Schieber, Quantum Symmetries for Exceptional SU(4) Modular Invariants Associ-ated with Conformal Embeddings, ArXiv:0805.4678.

[31] A. Ocneanu, Quantum Symmetries, Operator Algebras and Invariants of Manifolds, talk given at theFirst Caribbean Spring School of Mathematical and Theoretical Physics, Saint-Francois, Guadeloupe, 1993.


14/14

Dahmane Hammaoui


[32] V. B. Petkova and J. -B. Zuber, The Many Faces of Ocneanu Cells, Nucl. Phys. B 603 (2001), pp.449496.

[33] R. Coquereaux and R. Trinchero, On Quantum Symmetries of ADE Graphs, Adv. Theor. Math. Phys. 8(2004), pp. 189216.

[34] R. Coquereaux, The A2

Ocneanu Quantum Groupod, in Algebraic structures and their representations.Contemporary Mathematics 376 (2005), pp. 227247.

[35] R. Coquereaux, Racah-Wigner Quantum 6j Symbols, Ocneanu Cells for AN Diagrams and QuantumGroupods, J. Geom. Phys. 57(2) (2007), pp. 387434.

[36] R. Trinchero, Symmetries of Faces Models and the Double Triangle Algebra, Adv. Theor. Math. Phys.10 (2006), pp. 49-75.

[37] P. Di Francesco, P. Mathieu, and D. Senechal, Conformal Field Theory. New York: Springer-Verlag, 1997.

[38] E. Verlinde, Fusion Rules and Modular Transformations in 2D Conformal Field Theories, Nucl. Phys. B300 (1988), pp. 360376.

[39] J. Cardy, Boundary Conditions, Fusion Rules and the Verlinde Formula, Nucl. Phys. B 324 (1989), pp.

581596.

[40] E. Isasi: Methode de scission modulaire et symetries quantiques des graphes non-simplement laces entheorie des champs conforme, PhD Thesis, UP (Marseille), Oct. 2006.

[41] R. Coquereaux and A. Garcia, On Bialgebras Associated with Paths and Essential Paths on ADE Graphs,Int. J. Geom. Meth. Mod. Phys. 2 (2005), pp. 441466.

dahmane hammaoui- the smallest ocneanu quantum groupoid of su(3) type

Documents