数理解析研究所講究録1086 巻 1999 年 88-107

88-algebras and their applications
to reflection groups and conformal field theories$\mathrm{C}$J.-B. Zuber$CEA$

de Saclay,
SA CLAY Service de PhysiqueF-91191 Gif sur Yvette Cedex, France$Th\acute{e}or\dot{i}que$

The aim of this lecture is to present the concept of -algebra and to illustrate its applications in two contexts: the study of reflection groups and their folding on the one hand, thestructure of rational conformal field theories on the other. For simplicity the discussion$\mathrm{C}$current algebra,
is restricted to finite Coxeter groups and conformal theories with a.
but it may be extended to a larger class of groups and theories associated with$sl(2)\wedge$$sl(N)\wedge$

89

1. Introduction

The purpose of this talk is to present the notion of -algebra, a concept that appears
particularly suited in the discussion of various topics of current interest in mathematics
and mathematical physics: rational conformal field theories (rcft), topological field the$\mathrm{C}$ories, singularity theory and related problems. The concept was originally developed inrelation with finite groups and the algebras of their characters and classes (whence the”“): this exposes clearly one of the key features of these algebras, namely the pattern$\mathrm{C}$of two dual algebras. More generally, (the precise definition will be given in sect. 3),-algebras are associative, commutative algebras with a finite number of generators. They
come in dual pairs, endowed with different multiplication laws, one algebra being generated by the idempotents of the other. We shall illustrate and apply this concept in two$\mathrm{C}$different contexts: the association between rational conformal field theories and graphs on
the one hand; the folding of root systems and Dynkin diagrams on the other. In bothcases, generalized Dynkin diagrams are the central objects, and pairs of algebras that are
naturally associated with these graphs are -algebras. The study of the -subalgebras (to$\mathrm{C}$$\mathrm{C}$be also defined below) then enables one to understand the relationship between rcft andgraph-how to construct one object from the other-and to understand the folding of rootsystems, Dynkin diagrams and reflection groups.
Because certain positivity properties play an important role in the discussion of Calgebras, we start with a presentation of such properties that are empirically observed indifferent contexts but do not seem to have been given enough attention.
For the sake of brevity, all the discussion will be restricted to the simplest-and best
understood-case: rcft associated with $sl(2)$ , “minimal” topological field theories, simple

singularities, ordinary Coxeter-Dynkin diagrams, etc. There is ample evidence, however,-and a few proofs-, that the present considerations extend to a much larger context.

2. Three empirical facts,of one of the $ADE$ singularities. Here
Consider the prepotential
where is the rank of the associated $ADE$ algebra (the Milnor number of the singularity);satisfies thethe are the flat coordinates in the versal deformation of the singularity.
Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations [1], which express the associativity, and indices are, whereof the algebra with structure constants. The
raised and lowered with the -independent tensor $\eta_{kl}=C_{1kl}$ , and its inverse$\mathrm{t}=(t^{1}, \cdots , t^{n})$$F(\mathrm{t})$$n$$F(\mathrm{t})$$t^{j}$$C_{ij}k(\mathrm{t})$$t$$C_{ijk}( \mathrm{t})=\frac{\partial^{3}F}{\partial t^{t}\partial t^{j\partial}t^{k}}$$\eta^{jk}$90prepotential $F$ is a quasihomogeneous polynomial of degree $2(h+1)$ if we assign to thevariables
the degrees $\deg(t^{i})=(h+1-\lambda_{i})$ , whereis the i-th Coxeter exponentof the $ADE$ algebra: these exponents are supposed to be labelled in increasing order:$t^{i}$$\lambda_{i}$$\lambda_{1}=1<\lambda_{2}\leq\cdots\leq\lambda_{n-1}<\lambda_{n}=h-1,$

notations, labelling the$t’ \mathrm{s}$$h$the Coxeter number. It is convenient to changewith the value of, hence replacing$\lambda_{i}$$t^{i}$

with$t^{\lambda_{i}}$, and accordingly

to denote the structure constantsor, forexponents. The expressionsof the prepotentials for the various $ADE$ cases have been listed in the literature ([2-3] andfurther references therein).
Now, by inspection, we observe the following$C_{\lambda_{i}\lambda_{i^{\lambda}k}}$

Fact 1 : For the$A_{n},$$D_{2n},$$E_{6}$

and$E_{8}$$C_{\lambda\mu\nu}$cas es, th$\mathrm{e}re$$\lambda,$$\mu,$$\nu$

exists a choice of flat coordinat es forwhich all the coeffici ents of $F$ are real positi . For $D_{2n+1}$ an$\mathrm{v}e$$dE_{7}$th ere is no such choice.

Two remarks are in order.
First, why is it meaningful to look at reality and positivity properties in a problem thatlooks intrinsically complex?

Secondly it is curious that this splitting of the $ADE$ classification scheme into the sametwo sub-families appears also in other contexts. Let us quote
1) the structure of the modular invariant partition function of conformal field theories
with acurrent algebra. The latter are known to follow an $ADE$ classification
scheme $[4,5]$ . The question is to know if this partition function, which is a certain
sesquilinear form with non negative integer coeflicients, may or may not be written asa sum of blocks$sl(2)\wedge$$Z= \sum N_{\lambda\overline{\lambda}}\chi\lambda\overline{x}_{\overline{\lambda}}$$\Lambda_{\lambda\overline{\lambda}}’\in \mathbb{N}$$(2.1a)$$($$= \sum_{i}?|\sum_{\lambda\in\hat{\tau}_{\alpha}}\chi_{\lambda}|2$

For example, the cases labelled by$D_{10}$and$E_{7}$

2.1$b)$read respectively$Z^{(D_{1})}0=|x_{1}+x17|2+|x3+\chi 15|2+|x5+\chi_{1}3|^{2}+|x7+x11|^{2}+2|\chi 9|^{2}$$Z^{(E_{7})}=|\chi_{1}+\chi_{17}|2+|x5+x13|^{2}+|\chi_{7}+\chi_{11}|^{2}+|\chi_{9}|^{2}+$($(2.2a)$$(\chi_{3}+\chi_{15})x_{9}^{*}+\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e}\mathrm{X}$

conj.)$=|\chi_{1}+\chi 17|2|x5+\chi_{1}3|^{2}+|x7+x_{1}1|2++|x_{9}+\chi_{3}+\chi 15|2-|x_{3}+x_{1}5|2(2.2b)$(for more details and explanation of notations, see below sect. 5).
2) the positivity of the structure constants of the “Pasquier algebras” to be discussed

below;

913) the existence or non-existence of a “flat connection” on the path algebra on the Dynkin

diagram [6];4) the positivity of the coefficients of the prepotential just discussed;5) the positivity properties of the coefficients of the factors of the Poincar\’e polynomial of
the local (or “chiral”) ring of the singularity. Let us discuss briefly this latest aspect,
as it does not seem to be generally known. For any of the $ADE$ singularities, let. that enter the singulardenote the minimal number of non morsian variables$X_{i}$$p$polynomial. Let us write the Poincar\’e polynomial in the form
(2.3)

$P(t)= \prod_{i=1}^{p}\frac{(1-t^{h-}\deg(xi))}{(1-t^{\mathrm{d}\mathrm{e}}\mathrm{g}(X_{i}))}$

in terms of the degrees of the variables
degree of the singular polynomial.
$h$

$p$

$X_{i}$and of the Coxeter number , equal to the$h$exponents$\{\deg(xi)\}$$A_{n}$$n+1$

1$D_{l+2}$$2(P+1)$2

12,$E_{6}$

12

2

3,4

$\lambda$$1,2,$ $\cdots,$ $n$$2P+1,l+1$
1,4,5,7,8,11
1, 5, 7, 9, 11, 13, 17
1, 7, 11, 13, 17, 19, 23, 29

1, 3,

$l$

$\cdots,$

4,6
2
18
6,10
2
30
It is then an easy and amusing exercise to check that $P(t)$ may be written as a productof factors with positive coefficients only in the first subfamily. (I owe this observation
counterpart of the property 1)
to M. Bauer [7] . This is somehow the$E_{7}$$E_{8}$$p$$)$$\mathrm{m}\dot{\mathrm{u}}\mathrm{l}\mathrm{t}\mathrm{i}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}$mentionned above.
The interesting thing is that the simultaneous occurrence of several of these properties seem to extend beyond the $ADE$ case discussed here. The status of these various
occurences is however not the same. I think it is fair to say that 1) is the best understood,
as it is related to a structural property of the underlying conformal field theory. 2) isrelated to 4) as we shall see soon, but I doubt that 4) may be extended beyond the case of
simple singularities, as the prepotential is then no longer a polynomial. Finally it seems
that 5) does not generalize: for some singularities believed to be in correspondence withsome conformal field theory, property 5) may fail while 1) and 2) are true (for example,.
the singularity associated with the fusion potential of$sl(4)_{4}\wedge)$

In fact we are not going to make use of Fact 1 for generic , but only for a particularcase, obtained by the so-called Chebishev specialization. This refers to the deformation of$\mathrm{t}$92

the $ADE$ singularity for which all the flat coordinates but the one,$t^{n}$, with the smallest de-gree $\deg(t^{n})=2$ , (the largest exponent $\lambda_{n}=h-1$ ), i.e. the “less relevant” in the languageof physics, is kept non zero. As it is the only parameter in the homogeneous deformed polynomial $W(X_{1}, \cdots, X_{p}, tn)$ , one may rescale it to $t^{n}=1$ . The origin of the denomination isthat for the$A_{n}$

case, the deformed polynomial reads then $W_{A_{n}}(X_{1;}t^{n}=1)=T_{n+1}(x_{1})$ ,with $T_{n+1}(x)$ the degree $n+1$ Chebishev polynomial of first kind,$T_{n+1}(x)=2\cos(n+1),\theta$

if $x=2\cos\theta$ .

We also need some notations on the $ADE$ Dynkin diagrams. Let
denote theadjacency matrix of the Dynkin diagram under consideration:$b=1,$
label thevertices. The corresponding Cartan matrix is $C_{ab}=2\delta_{ab}-G_{ab}$ . The eigenvectors$G_{ab}$$a,$$\cdots,$ $n$$\psi^{(\lambda)}$and eigenvalues of these symmetric matrices are indexed by the Coxeter exponents ,$\lambda$$G_{ab} \psi^{(\lambda)}b=2\cos\frac{\pi\lambda}{h}\psi^{(\lambda)}a$The$\psi^{(\lambda)}$(2.4)

may be chosen orthonormal.

Then we can state theFact 2 :$di$

The structure const anagon alized by the$ts$of the chiral ring in the Chebishev speci alization$\mathrm{a}\mathrm{r}e$$\psi^{(\lambda)}$$M_{\lambda\mu} \nu:=C_{\lambda\mu}\mathcal{U}(t^{n}=1)=\sum_{a}\frac{\psi_{a}^{(\lambda)()}\psi a\mu\psi_{a}*(\nu)}{\psi_{a}^{(1)}}$(2.5)

Here I have introduced the notation $M$ to be used in the forthcoming discussion. In
the denominator of the right hand side, there appears the exponent 1, that yields thelargest eigenvalue of the matrixof$\psi^{(1)}$$G$. By the Perron-Frobenius theorem, all the components

are non vanishing and of the same sign.

Fact 2 is not a surprise in the$A_{n}$

case, where it follows from the combined workof Verlinde [8] and Gepner [9]. Indeed the above structure constants reduce then to thefusion coefficients of the$sl(2)\wedge$

algebra, for a value of the level (central extension) equal to$k=n-1$ , and the latter are known to have an interpretation in terms of the chiral ringof a topological field theory. For the other $D$ and $E$ cases, the observation was made (inessence, not quite in these terms) by Lerche and Warner [10], and made more systematicand extended in [3].

93

The previous formula suggests to consider also the dual algebra (we shall see belowthat the word “dual” is legitimate), with structure constants(2.6)$N_{ab}C:= \sum_{\lambda}\frac{\psi_{a}\psi(\lambda)(\lambda)b\psi_{C}^{(}\lambda)*}{\psi_{1}^{(\lambda)}}$where the sum runs over the exponents of the case at hand. This definition depends
are non-vanishing. Such a vertex
on a choice of a vertex denoted 1 for which all theexists for all the $ADE$ cases. There may remain, however, some arbitrariness in the choice
for which an exponent occurs with multiplicityof that vertex 1 and also, in the case$\lambda$$\psi_{1}^{(\lambda)}$$D_{2n}$2, in the choice of the basis
Fact 3 :For the$A_{n},$$\psi_{a}^{(\lambda)}$$D_{2n},$$E_{6}$. Now comes theand$E_{8}$

cases, thconst ants
such that the struct, there exists no such choice.
cases $D_{2n+1}$ and
$b$

asis

$\psi_{a}^{(\lambda)}$$\mathrm{u}re$exists a choice of vertex 1 and of theare all non negative. For theand$\mathrm{e}re$$M_{\lambda\mu}\nu$$N_{ab}c$$E_{7}$

Fact 2.
Note that the non-negativity of the $M$ is a simple consequence of Fact 1
For the $ADE$ cases, the numbers $N$ turn out to be integers (with an adequate choice of$\cap$

1 and the basis). The interpretation of these numbers in the various contexts in which
they occur (conformal field theories, topological theories and singularities, lattice models)has remained elusive so far. In contrast, the $M$ that are in general non integers but

rather algebraic numbers, have such an interpretation: they give the structure constantsof the chiral ring of the Chebishev specialization, as just explained; in the context ofconformal field theories and integrable lattice models, they give the coupling constants offield operators [11], [8], [12]. It is in that context that this algebra was first introduced by
Pasquier [11], whence the name of Pasquier algebras that I give to the pair of $M$ and $N$

algebras.

3. C-algebras

3.1. Definitions and examples”
The appropriate language to discuss these Pasquier algebras is that of -algebras, (“by Kawada and recently reviewed and revived byfor character), introduced in the$C$$\mathrm{C}$$40’ \mathrm{s}$Bannai and Ito [13].

ut overDefinition : An alge
satisfi es the following axioms:$\mathrm{b}r\mathrm{a}$$\mathbb{C}$with a given

$b$

asis

$x_{1},$$\cdots,$ $x_{n}$, is a$C$-alge$\mathrm{b}ra$

if

$it$

94

$i)$

it is a commutative and associative algebra with real structure constants$Xa \cdot xb=\sum_{C}p_{ab}cx_{c}$$\mathrm{i}i)$$p_{ab^{C}},$

$i.e$

.

;

it has an identity element, denoted$x_{1},$$i.e$.

$p_{1a}=\delta bab$;there is an involu tion on the gen era tors
that is an automorphism of thealgebra, $i.e$ .
;, witha real positive number $k_{a}>0$ ;v) theform a one-dimensional representation of the algebra.
Among the various consequences of these axioms, is the fact that is semi-simple.
There are one-dimensional representations of the algebra, that we label by an index
taking values:. The valuerefers to the special representation of$p_{a}(1)=k_{a}$axiom v):. Ifdenote the corresponding idempotents, one may decompose
The matrix
is invertible, letdenote the matrix suchthat
. More explicitly, the matricesof elements
form a representation of the algebra . They are normal and commuting,and thus diagonalizable in a common orthonormal basis
. All
and
are nonvanishing and may thus be chosen real positive. One may write$iii)$$x_{a}-+x_{\overline{a}}$$p_{ab}=p_{\overline{a}\overline{b}}c\overline{C}$$iv)p_{ab}1=k_{a}\delta_{a\overline{b}}$$k_{a}$$k_{a}$$\mathfrak{U}$$n$$\lambda$$n$$x_{a}\mapsto p_{a}(\lambda)\in \mathbb{C}$$\lambda=1$$e_{\lambda}$$x_{a}= \sum_{\lambda}p_{a}(\lambda)e_{\lambda}$$p_{a}(\lambda)$$\sum_{\lambda}p_{a}(\lambda)q_{\lambda}(b)=\kappa\delta_{ab},$$q_{\lambda}(a)$$\kappa:=\sum_{a}k_{a}$$P_{a}$$(P_{a})_{b}^{c}=\sqrt{\frac{k_{c}}{k_{b}}}p_{ab^{C}}$$\mathfrak{U}$$\psi_{a}^{(\lambda)}$$\psi_{1}^{(\lambda)}$$\psi_{a}^{(1)}$$p_{ab}=c \sqrt{\frac{k_{a}k_{b}}{k_{c}}}\sum\frac{\psi_{a}^{(\lambda)}\psi^{(}b\psi_{\mathrm{C}}\lambda)(\lambda)*}{\psi_{1}^{(\lambda)}}\lambda$$\sqrt{k_{a}}=\frac{\psi_{a}^{(1)}}{\psi_{1}^{(1)}}$(3.1)$p_{a}( \lambda)=\frac{\psi_{a}^{(\lambda)(1}\psi_{a})}{\psi_{1}^{(\lambda)}\psi^{()}11}$$q_{\lambda}(a)= \frac{\psi_{a}*\psi_{1}(\lambda)(\lambda)}{\psi_{a}^{(1)}\psi^{()}11}$

and let$\hat{k}_{\lambda}$

be such that
(3.2)$\sqrt{\hat{k}_{\lambda}}=\frac{\psi_{1}^{(\lambda)}}{\psi_{1}^{(1)}}$

One may then show that the dual of Ut, defined as the set of linear maps from into ,is endowed with a structure of -algebra: its basis is labelled by the , its one
dimensional
representations are provided by the, among which
are positive, and thestructure constants of the algebra are$\hat{\mathfrak{U}}$$\mathfrak{U}$$\mathbb{C}$$\mathrm{C}$$\lambda$$q_{\lambda}(a)$$q_{\lambda}(1)=\hat{k}_{\lambda}$$q_{\lambda\mu}= \nu\sqrt{\frac{\hat{k}_{\lambda}\hat{k}_{\mu}}{\hat{k}_{\nu}}}\sum_{a}\frac{\psi_{a}^{(\lambda)}\psi^{(\mu)()}a\psi a*\nu}{\psi_{a}^{(1)}}$

(3.3)

95

The

$k_{a}$

and$\hat{k}_{\lambda}$

are called the Krein parameters of the algebras. They satisfy $\kappa=\sum_{a}k_{a}=$$\sum_{\lambda}\hat{k}_{\lambda}=1/\psi_{1}(1)2$

Alternatively, one may regard this dual as a second -algebra structure on Ut, withis endowed with a pair of dual C. To recapitulate,and idempotents
basis
in the basis , andalgebra structures, one with multiplication., structure constants$\hat{\mathfrak{U}}$$\mathrm{C}$$\mathfrak{U}$$x_{a}$$\kappa e_{\lambda}$$p_{ab}C$

idempotents$\kappa e_{\lambda}$, and the other with multiplication , structure constants$0$$e_{\lambda}$

and idempotents$x_{a}$$q_{\lambda\mu}\nu$in the basis$x_{a}$$x_{a}.x_{b}= \sum_{C}p_{ab}x_{c}C$,$e_{\lambda}.e_{\mu}=\delta_{\lambda\mu}e\lambda$(3.4)$\kappa e_{\lambda^{\circ\kappa e}\mu}=\sum q_{\lambda\mu}\nu\nu\kappa e_{\mathcal{U}}$,$x_{a}\mathrm{o}x_{bb^{X_{a}}}=\delta a$

Examples:denote its
1. Character and class algebras of a finite group. Let be a finite group,
the value
their characters,
its irreducible representations,equivalence classes,of these characters on class $a;a=1$ refers to the class of the identity, $\rho=1$ to the identity
is the dimension of representation . One has two dual algebras
representation;$C_{a}$$\Gamma$$\chi_{a}^{(\rho)}$$\chi^{(\rho)}$$(\rho)$$d_{\rho}=\chi_{1}^{(\rho)}$$\rho$$C_{a}C_{b}=cbCacC$$–(3.5)$$xx=(\lambda)(\mu).I\mathrm{t}\chi(’\lambda\mu\nu\nu)$Introducing theorthonorma! by virtue of the standard orthogonality andcompleteness relations of characters, one may write$\chi_{a}^{\lambda}\wedge=\sqrt{\frac{|C_{a}|}{|\Gamma|}}\chi_{a}^{(\lambda)},$$p_{ab}=C_{ab}cc= \sqrt{\frac{|C_{a}||Cb|}{|.C_{C}|}}\sum$$\frac{\chi_{a}^{(\lambda)}x_{b}\chi^{(\lambda)*}\wedge\wedge(\lambda)\wedge C}{\chi_{1}^{(\lambda\rangle}\wedge}\lambda$

(3.6)

.

$q_{\lambda\mu} \nu=\frac{d_{\lambda}d_{\mu}}{d_{\nu}}K^{\lambda\mu}\nu=\frac{d_{\lambda}d_{\mu}}{d_{\nu}}\sum_{a}\frac{\chi_{a}x\wedge(\lambda)\wedge(a\mu)\chi_{a}^{(}\wedge\nu)*}{\chi_{a}^{(1)}\wedge}$

The two dual algebras have integer Krein parameters.
known relation$k_{a}=|C_{a}|,\hat{k}_{\lambda}=d_{\lambda}^{2}$

with the well$| \Gamma|=\sum k_{a}=\sum\hat{k}_{\lambda}=\sum d_{\lambda}^{2}$

2. The Pasquier algebras introduced above are obviously a pair of dual -algebras. The, as indiand
are respectively proportional toandstructure constants
cated in (3.1) and (3.3). In that case, in contrast with example 1, the Krein parameters
are not integers. Among these Pasquier algebras, there are the fusion algebras of affine$\mathrm{C}$$p_{ab}C$$q_{\lambda\mu}\nu$$N_{ab}c$$M_{\lambda\mu}\nu$96algebras . In that case, the two dual algebras are in fact isomorphic: this is due to the fact$\hat{\mathrm{g}}$that according to the Verlinde formula, the diagonalizing matrix is the symmetric unitarymatrixof modular transformations of the affine characters [8]. Also, in that case, theKrein parameters are equal to, that is, the square of the quantum$S$$\hat{k}_{\lambda}=(\frac{s_{1\lambda}}{s_{11}})^{2}$$\hat{k}_{\lambda}=D_{\lambda}^{2}$dimension of the corresponding representation of . This is thus a quantum deformationof the finite group situation of the previous example.
$\hat{\mathrm{B}}$3.2. C-subalgebras
One then defines$\mathrm{C}$-subalgebras of a C-algebra:

Definition : Given aalge$br\mathrm{a}$$i.\mathrm{e}.$

if

$a,$

$C$

-algebra with a basis

genera $ted$ by a subset of the$b\in T,$ $p_{ab}c\neq 0$$x_{a},$$a=1,$$\cdots,$ $n$$a\in T,$ $T\subset\{1, \cdots, n\}$

, a

$C$

-subalgebra is a C-

, closed under multiplication,

only if $c\in T$ .

Note that this condition implies that$a\mapsto\overline{a}[13]$$\{x_{a}\},$$T$

contains 1 and is stable under the involution.

We shall be mainly interested in the situation where the two dual algebras have nonnegative structure constants. Then there is a remarkable theorem that tells us that theexistence of a -subalgebra in$\mathrm{C}$precisely, suppose$\mathfrak{U}$has a$\mathrm{C}$$\mathfrak{U}$implies the existence of a

-subalgebra

define an equivalence relation $a\sim b$ if$X_{1}= \sum_{T}x_{a}$. One also defines

into idempotents$\mathrm{t}\mathrm{h}\mathrm{e}_{\vee}$$T$. One may then

: $p_{ac}b\neq 0$ , and there is a partition of the$T_{i},\dot{i}=1,$ $\cdots,p,$subset-subalgebra in the dual. Moreassociated with a subset$\exists c\in T$

set $\{1, 2, \cdots n\}$ into equivalence classes,$X_{i}:= \sum a\in\tau_{i}xa$$\mathfrak{U}_{T}$$\mathrm{C}$. Let$\rho=\sum_{a\in T}k_{a}$and letof the dual basis by the decomposition of$\hat{T}$$X_{1}= \rho\sum_{\lambda\in\hat{T}}e_{\lambda}$$T_{1}\equiv T$.

Theorem (Bannai-Ito [13], theorem 9.9): Consid er a -algebra ut with non nega
stru cture constants
and. With the notations just introduced,$C$$p_{ab}C$$i)t \mathrm{A}e\frac{1}{\rho}X_{i},$$\mathfrak{U}/\mathfrak{U}_{T}$$i=1,$$\cdots,\cdot p,$$q_{\lambda\mu}\nu$

$g$

enera

$te$

th

$\mathrm{e}m\mathrm{s}$, with a product $inh$ erited from

$ii)$

the

$iii)$

these two$\kappa e_{\lambda}$

, for
$C$

$\lambda\in\hat{T}$, genera-algebras are$t\mathrm{e}$elves aa C-su algebra$d\mathrm{u}al$quotients$\mathfrak{U}/\mathfrak{U}_{T}$

and

$C$

-alge

$br\mathrm{a}$, call$\mathrm{e}d$the quotient C-algebra$\mathfrak{U},\cdot$$b$$\hat{\mathfrak{U}}_{\hat{T}}$of the dual algebra$\hat{\mathfrak{U}}$;to one another.

Thus one has a dual pattern of subsets$\hat{\mathfrak{U}}/\hat{\mathfrak{U}}_{\hat{T}}$$ti\mathrm{v}e$

$T$

and

with the isomorphisms$\hat{T}$, of -subalgebras$\mathrm{C}$$\overline{\mathfrak{U}/\mathfrak{U}_{T}}\cong\hat{\mathfrak{U}}_{\overline{T}}$$\mathfrak{U}_{T}$andand vice versa$\hat{\mathfrak{U}}_{\hat{T}}$, and of$\hat{\mathfrak{U}}/\hat{\mathfrak{U}}_{\hat{T}}\cong\overline{\mathfrak{U}_{T}}$.

97, and conversely.
may be expanded on the
One proves also that all
andexpressions ofand’ withRecalling thatgiven in (3.1), we find that$\lambda\in\hat{T}$$X_{i}$$e_{\lambda},$$p_{a}(\lambda)$$\kappa e_{\lambda}=\sum_{a}q_{\lambda}(a)Xa$$x_{a}= \sum_{\lambda}p_{a}(\lambda)e_{\lambda}$$q_{\lambda}(a)$$\sum_{a\in T_{i}}p_{a}(\lambda)=0$

thus
for

$\sum_{a\in T_{i}}\psi^{(\lambda)}a\psi(a1)=0$$\lambda\in\hat{T}$if$\lambda\not\in\hat{T}$if$\lambda\not\in\hat{T}$$q_{\lambda}(a)= \frac{\psi_{a}^{(\lambda)*}}{\psi_{a}^{(1)}}\frac{\psi_{1}^{(\lambda)}}{\psi_{1}^{(1)}}$(3.7)

independent of.

$a\in T_{i}$These two conditions may be conveniently assembled into a single one$\forall\lambda,\forall T_{i},$$\forall a\in\tau_{i}$$\sum_{b\in Ti}\psi_{b}\psi_{b}^{(1})(\lambda)=\delta_{\lambda\in\hat{T}^{\frac{\psi_{a}^{(\lambda)}}{\psi_{a}^{(1)}}}}\sum_{b\in Ti}(\psi_{b}^{()})^{2}1$

(3.8)

,

a form that will be useful in the sequel. It is also easy to write explicitly the expressionsof the structure constants of the quotient algebras. For example, from $X_{i}= \sum_{a\in T_{i}}Xa$ it
withfollows that$\frac{1}{\rho}X_{i}.\frac{1}{\rho}X_{j}=\sum_{k}\mathrm{p}_{ij^{k}}\frac{1}{\rho}X_{k}$$\mathrm{p}_{ij}k=\frac{1}{\rho}\sum_{c\in Tk}pabC,$$\forall a\in T_{i},$$b\in T_{j}$

.

(3.9)

In the following two sections, I shall present two applications of this theorem. The
first deals with reflection groups and their folding, the second with conformal field theories.
The first starts with -subalgebras of the $M$ algebra (subject to an additional constraint),$\mathrm{C}$the second with those of the $N$ algebra.
4. Folding of $ADE$ Dynkin diagrams

4.1. The problem) may be
It is well known that non simply laced Dynkin diagrams (of typeobtained by folding the simply laced ones, using the symmetries of the original diagram.
The extension to Coxeter diagrams of $H$ or type, associated with the non-crystallographic
groups, seems more recent $[14,15]$ . In all these works, one is given a simply laced$B_{n},$ $c_{n},$ $F_{4},$ $G_{2}$$I$$\mathrm{c}_{\mathrm{o}\mathrm{X}\mathrm{e}}\mathrm{t}\mathrm{e}\mathrm{r}$Dynkin diagram describing the scalar products of a set of simple roots
according to$(\alpha_{a}, \alpha_{b})=2\delta ab-G_{ab}$$\{\alpha_{a}\},$$a=1,$$\cdots$

, ,
$n$

(4.1)

98

the adjacency matrix as in (2.4)). Then a certain partition is found of this set into
subsetsof mutually orthogonal roots($G$$\{\alpha_{a}, a\in T_{i}\}$if$(\alpha_{a}, \alpha_{b})=0$

Let$S_{a}$$a,$ $b\in T_{i}$.

denote the reflection in the hyperplane orthogonal to

the group generated by all the$S_{a},$$a=1,$$\cdots,$ $n$(4.2)

through the origin,. Then one forms the products(4.3)$R_{i}= \prod_{a\in T_{i}}s_{a}$in which the order is immaterial, since the$\mathrm{a}\mathrm{n}\dot{\mathrm{d}}G$$\alpha_{a}$are orthogonal within the same , and thus
thecommute. The group $G’$ generated by theis clearly a subgroup of . Since
is a Coxeter group (of finite order), $G’$ is also of finite order, hence in the $A-I$ list.
The corresponding Coxeter diagram thus results from identifying the vertices of a same
block , while the superscript of an edge $i-j$ , which yields the order of the element
may be computed easily in terms of the original. One finds empirically the adequate
foldings of the $A,$ $D,$ $E$ diagrams necessary to manufacture all the others (see Fig. 1). For
example, the order 5 of the productin the diagram. the smallest power $m$. $(S_{2}S_{3}S4S_{6})^{m}=I$ is simply the order of the Coxeter element of the
Coxeter group
generated by these four reflections.
As far as I can see, this procedure is, however, empiric, and doesn’t say which folding
does the job and in which subspace of the original -dimensional space the subgroup acts.
In the fairly different context of topological field theories, a parallel observation was
made. Starting from the so-called minimallabelled by $ADE$ Dynkin diagrams, i.e.
solutions of the WDVV equations of the type mentionned in sect. 2, one finds that there
are other solutions obtained by restriction of the latter. In such a restriction, only a subsetof the flat coordinates is kept non-vanishing. These non-vanishing
are labeled by the
Coxeter exponents of some non simply laced Coxeter groups $[16,17]$ . These restrictions are
consistent with the algebra of the , in the sense that they correspond to a sub-algebra ofthe. If we consider the Chebishev specialization and recall Fact 2 of sect. 2, thismeans that the Pasquier algebra $M$ of the original $ADE$ diagram admits a sub-algebra,
whose generators are labelled by the exponents of a Coxeter group of type $B,$ $C,$ $F,$ $G-I$$T_{i}$$\alpha$$S_{a}$$G$$R_{i}$$G$$T_{i}$$R_{i}R_{j}$$S_{a}$$R_{2}R_{3}$$H_{3},\dot{i}.e$$\mathrm{s}.\mathrm{t}$$A_{4}$$n$$(\mathrm{t}\mathrm{f}\mathrm{t})$$\mathrm{t}\mathrm{f}\mathrm{t}’ \mathrm{s}$$t$$t’ \mathrm{s}$$\mathrm{t}\mathrm{f}\mathrm{t}$$C_{\lambda\mu}\nu(\mathrm{t})$[17].

In fact there is a strong connection between the two observations, and through thetheory of -algebras, one is able to answer the previous objection and determine the folding
through the study of the subalgebras of $M$ type.
$\mathrm{C}$99

4.2. From a$M$-subalgebra to a subgroup

Consider a simply laced $ADE$ Dynkin diagram such that the structure constants $M$ and$N$ are non negative (see Fact 3 of sect.
2). Recall that all Dynkin diagrams may be. their vertices may be assignend a
grading , the “colour”, such that
2-coloured,$G_{ab}--0$ if $\tau(a)=\tau(b)$ . Now suppose that a subalgebra of the $M$ algebra has been found,of exponents such that. a subset$\dot{i}.e$$\mathbb{Z}_{2}$$\tau$$\hat{T}$$\dot{i}.e$$\lambda,$$\mu\in\hat{T}$$M_{\lambda\mu}\nu\neq 0\Rightarrow\nu\in\hat{T}$

(4.4)

;

. The positivity
condition tells us that we are in the conditions of the theorem of sect. 3.2. Because here
we start from a -subalgebra of the $M$ (or ) algebra, the theorem has to be transposed
to its dual version, namely,(i) there is a partition of the set of exponents into equivalence classes
the subsetof exponents is assumed to be stable under$\hat{T}$$\lambda\mapsto h-\lambda$$\mathrm{C}$$q$$\hat{T}_{\alpha}$if$\mu\sim\nu$$T$$N_{ab}C\neq 0$$M_{\lambda\mu}\nu\neq 0$

;

(4.5)

of the dual set of vertices that contains 1;
enables one to define a dual equivalence relation: $b\sim c$ if(ii) there exists a special subset(iii) the set$\exists\lambda\in\hat{T},$$T$$\exists a\in T$such that, and hence a partition of the set of vertices into equivalence classes$T_{i;}$(iv) the relation (3.8) is satisfied.

implies that:is stable under; (ii) the class contains only vertices satisfying(i) the same is true for each classhave the same
more generally all the vertices within a same classand thus the corresponding roots are mutually orthogonal. These are trivial
colour
Now the assumption made above that$\lambda\mapsto h-\lambda$$\hat{T}$$T$$\hat{T}_{\alpha}$$a$$T_{i}$$\tau(a)=\tau(1))(\mathrm{i}\mathrm{i}\mathrm{i})$$\tau$

consequences of the symmetry of the$\psi$(4.6)$\psi_{a}^{(h-\lambda)}=(-1)^{\mathcal{T}()}a\psi_{a}^{(}\lambda)$$G$

I now claim that with this pattern of subalgebras one may associate a subgroup of
correspondence with; it is again described by a graph, whose vertices are in$\mathrm{o}\mathrm{n}\mathrm{e}- \mathrm{t}_{\mathrm{o}^{-}\mathrm{o}\mathrm{n}\mathrm{e}}$and whose set of exponents is . This subgroup is generated by reflectionsin the hyperplanes orthogonal to some , that are some linear (real) combinations of thethe classes$\hat{T}$$T_{i}$$\beta$roots$\alpha$:$\beta_{i}=N_{i}\sum_{\tau a\in i}\psi_{a}(1)a\alpha$

;

(4.7)

100

the normalisation is adjusted so that$(\beta_{i}, \beta_{i})=2$, namely(4.8)$N_{i}^{2} \sum_{a\in Ti}(\psi_{a}^{(1)})2=1$(since the$\prod_{a\in}\tau_{i}S_{a}$$\alpha_{a},$$a\in T_{i}$are mutually orthogonal). One verifies, using (3.8), that the producthas the same action as the reflection

the subspace spanned by the$\beta[18]$$R_{i}$

in the hyperplane orthogonal to$\beta_{i}$

, in

.

The scalar products of two distinct roots$\beta_{i}$andsame property for the original simple positive roots

is non positive, as follows from the$\beta_{j}$$\alpha_{a}$and from the positivity of the$\psi_{a}^{(1)}$.
$( \beta_{i}, \beta_{j})=N_{i}N_{j}b\in T\sum_{a\in T_{ij}}(\alpha_{a}, \beta_{b})\psi_{a}(1)\psi_{b}(1)\leq 0$

The metric defined on the original roots may be diagonalized by the$g_{ab}=( \alpha_{ab}, \alpha)=\mathrm{o}\mathrm{n}\sum_{\exp \mathrm{e}\mathrm{n}\mathrm{t}\mathrm{s}\lambda}g\psi_{a}\lambda)\psi_{b}^{(}(\lambda)(\lambda)*$with$g^{(\lambda)}=2-2 \cos\frac{\pi\lambda}{h}$. From the expressions of the new roots$\psi$,$\beta_{i}$(4.9)

it is easy to computethe new metric, making use again of (3.8)$\forall a\in T_{i},$ $\forall b\in T_{j}$$g_{ij}=( \beta i, \beta_{j})=\Lambda_{i^{-}}^{\prime 1}N_{j}^{-}1\sum g^{(}\lambda\in^{\hat{\tau}}\lambda)_{\frac{\psi_{a}^{(\lambda)}}{\psi_{a}^{(1)}}}\frac{\psi_{b}^{(\lambda)*}}{\psi_{b}^{(1)}}$(4.10)$= \sum_{\lambda\in\hat{T}}g\Psi^{(}(\lambda)\lambda)\Psi_{j}(\lambda)*i$in terms of the new eigenvectors$\lambda\in\hat{T}$$\Psi_{i}^{(\lambda)}=N_{i}^{-1}\frac{\psi_{a}^{(\lambda)}}{\psi_{a}^{(1)}}$$\forall a\in T_{i}$(4.11)$= \Lambda_{i}’\sum_{a\in\tau i}\psi_{a}(\lambda)\psi(1)a$These eigenvectors form an orthonormal system of rank$|\hat{T}|$

.

101

$\mathrm{E}\mathrm{x}\mathrm{p}=\mathrm{t}\mathrm{l},2,$$\ldots,\mathrm{k}+1\mathrm{I}$or$\mathrm{I}(\mathrm{k}+2)2\underline{\mathrm{k}+2}$$\underline{\mathrm{k}+2}$$\wedge \mathrm{T}=\{1, \mathrm{k}+1\}$$\mathrm{E}\mathrm{x}\mathrm{p}=\{1,2,$$\mathrm{B}_{\mathrm{k}+1}\ovalbox{\tt\small REJECT}^{4}12\mathrm{k}+1$$\mathrm{D}_{4}$$\ldots,2\mathrm{k}+1\mathrm{I}$$\wedge \mathrm{T}=\{1,3, \ldots,2\mathrm{k}+1\}$$\mathrm{E}\mathrm{x}\mathrm{p}=\{1,3,5,3\}$$\underline{6}$$\wedge \mathrm{T}=\{1,5\mathrm{I}$$\mathrm{G}_{2}$$\mathrm{D}_{6}$$\mathrm{H}_{3}$$\mathrm{E}\mathrm{x}\mathrm{p}=\{1,3,5,7,9,5\}$$\underline{5}$$\wedge \mathrm{T}=\{1,5,9\}$$\mathrm{E}_{8}$$\mathrm{E}\mathrm{x}_{\mathrm{P}^{=}\mathrm{t}\}}1,7,11,13,17,19,23,29$$\mathrm{H}_{4}$$\underline{5}$$\mathrm{E}\mathrm{x}\mathrm{p}=\{1,4,5,7,8,11\}$$\mathrm{E}_{6}$$\mathrm{F}_{4}$$\wedge \mathrm{T}=i1,11,19,29\}$$\underline{4}$$\wedge \mathrm{T}=\{1,5,7,11\}$

Fig. 1: The folding of ADE Dynkin diagrams of positive type. Classes
vertices encompass nodes on the same vertical.

$T_{i}$

of102

4.3. Discussion
The reader may wonder what happens in (4.7) if the Perron-Frobenius eigenvector$\psi^{(1)}$ischanged into another eigenvector. In fact, this has the effect of giving roots of the folded
diagram that are simple but not positive.
The result of the procedure is presented in fig. 1. For each simply laced Dynkin

diagram of type $A,$

$D_{2\ell},$$E_{6}$

satisfying the invariance ofor$\hat{T}$$E_{8}$, a systematic search of subalgebra of the $M$ algebra,

under$\lambda\mapsto h-\lambda$

has been carried out. All cases are notexhibited in the Figures, as there is some redundancy. For example, any diagram of theprevious type admits a subalgebra associated with $\hat{T}=\{1, h-1\}$ . This corresponds tofolding all vertices of a given colour onto one another, resulting in a 2-vertex graph of type
$I_{2}(h)$. This has been represented only for $A_{k+1}\mapsto I_{2}(k+2)$ or $D_{4}\mapsto G_{2}\equiv I_{2}(6)$ .

$\mathrm{E}\mathrm{x}\mathrm{p}=\mathrm{t}1,3,$$\ldots,$$\mathrm{Q}_{\mathrm{p}+1}\overline{12}$. ..

$\frac{4}{2\mathrm{p}2\mathrm{p}}+1$$4_{\mathrm{P}}+1,2\mathrm{p}+1\}$$\wedge \mathrm{T}=\mathrm{t}1,3,$$4\mathrm{p}+1$$\ldots,$}

Fig. 2: A case of folding which is discarded by the assumption of positivity
By inspection of fig. 1, the reader may convince herself or himself that the procedure

is exhaustive, in the sense that all non simply laced Coxeter diagrams, or all Coxeter

groups, have been obtained. In fact, one possible folding of $D_{2p+2}$ into $C_{2p+1}$ (fig. 2) does
not appear in the present discussion. To expose the corresponding $M$ subalgebra of the$D_{2p+2}$

diagram requires indeed to change the basis of eigenvectors$\psi$

into another one, inwhich positivity is lost [12]. In the present case, because of the isomorphism of$C_{n}$$B_{n}$

and

Coxeter-Weyl groups, this does not hinder the exhaustivity, but we may expect thatthe extension of the method to more general cases may require relaxing the hypothesis of

positivity. We refer the reader to [18] for a discussion of the appropriate extension of thepresent method.

103

5. Dynkin diagrams and RCFT

I shall be more concise on this part as it has already been expounded elsewhere $[12,19]$ .

As recalled above in sect 2, conformal field theories with a$sl(2)\wedge$

current algebra have beenclassified according to an $ADE$ scheme. This manifests itself first in the form of theirmodular invariant genus 1 partition function, written as a sesquilinear form of characters$\chi_{\lambda}(q),$$\lambda$$q=e^{2i\pi\tau}$, of the affine

algebra at a given level , with the integrable weights$sl(2)\wedge$$k$labelled by integers $1\leq\lambda\leq k+1$ . One proves [4-5] that the possible expressions of thatpartition function$z= \sum N_{\lambda\overline{\lambda}\chi\lambda(}q)\overline{x}\overline{\lambda}(\overline{q})$are such that the $d_{i}ag,onal$ terms$\lambda=\overline{\lambda}$(5.1)$N_{\lambda\overline{\lambda}}\in \mathbb{N}$are the Coxeter exponents of one of the $ADE$

Dynkin diagrams of Coxeter number $h=k+2$ .

As alluded to in sect 2, the $A,$$Z$

is a sum of blocks$D_{2f},$$E_{6}$$Z= \sum_{\alpha}|\sum_{\lambda\in\overline{T}_{\alpha}}\chi_{\lambda}|^{2}$and$E_{8}$

cases-and only those-are such that:$k+2=n+1$ ,$Z^{(A_{n})}= \sum_{1\lambda=}^{n}|\chi_{\lambda}|2$$Z^{(D_{2t})}=$$k+2=4\ell_{-2}$$|\chi_{\lambda}+\chi_{4}l-2-\lambda|2+2|x_{2l}-1|2$$\sum$$\lambda=1,3,\cdots,2\ell-3$$Z^{(E_{6})}=|\chi_{1}+\chi 7|2|+x4+x_{8}|2+|\chi_{5}+x11|^{2}$$k+2=12$$Z^{(E_{8})}=|\chi_{1}+\chi 11+x19+x_{2}9|^{2}+|\chi 7+\chi_{13}+\chi 17+\chi 23|^{2}$$k+2=30$ .

(5.2)

This pattern reflects the existence of an underlying “extended” chiral algebra, containing
the current algebra$sl(2)\wedge$

as a subalgebra. The combinations$\hat{\chi}_{\alpha}=\sum_{\lambda\in\hat{T}_{\alpha}}\chi\lambda$

that appearin (5.2) are characters of the extended algebra decomposed into irreducible characters of$sl(2)\wedge$. Let us denote$S_{\lambda\mu}$, resp$\mathrm{S}_{\alpha\beta}$, the matrices of modular transformations of the two

sets of characters$x \lambda(\tilde{q})=\sum_{\mu}s\lambda\mu\chi\mu(q)$$\hat{\chi}_{\alpha}(\tilde{q})=\sum_{\beta}\mathrm{s}_{\alpha}\beta\hat{x}_{\beta(q})$

where$\tilde{q}=e^{\frac{-2i\pi}{\tau}}$

One has$\mathrm{S}_{\alpha\beta}=\sum_{\lambda\in\hat{T}_{\alpha}}s_{\lambda}\mu’\forall\mu\in\hat{T}_{\beta}$

representations are the ratios$\mathrm{D}_{\alpha}=\mathrm{S}_{\alpha 1}/\mathrm{S}_{\mathrm{l}1}$

and

(5.3)

,

. The quantum dimensions of the$D_{\lambda}=S_{\lambda 1}/S_{11}$

.

104

It has been observed in $[12,19]$ that there is a second manifestation of the $ADE$

diagrams hidden in the structure of the operator algebra. For the theories (5.2), one
proves that the fusion coefficientsof the extended algebra satisfy$\mathrm{N}_{\alpha\beta}\gamma$$\mathrm{N}_{\alpha\beta}\gamma=\sqrt{\frac{\mathrm{D}_{\alpha}}{D_{\lambda}}}\sqrt{\frac{\mathrm{D}_{\beta}}{D_{\mu}}}\sum_{\mathcal{U}\in\hat{T}_{\gamma}}M\nu\sqrt{\frac{D_{\nu}}{\mathrm{D}_{\gamma}}}\lambda\mu$$\mu\in\hat{T}_{\beta}.$’$\forall.\lambda\in\hat{T}_{\alpha},$

,

(5.4)

where $M$ are the structure constants of the Pasquier algebra of the relevant Dynkin dia-gram. (For the sake of simplicity, we assume here and in the rest of the discussion thatnone of the exponents has a multiplicity larger than 1: this excludes thecase. The
cases with multiplicities require a more elaborated labelling, see [19] .
This equation has several interesting consequences. First, since the matrixisdiagonalized by thematrix, according to the Verlinde formula, it follows from (5.4)
that, whereis the blo.ck containing
and is any representationof the extended algebra, forms a one-dimensional representation of the $M$ algebra, i.e.
, and may thus be identified with some
, for some vertex$D_{\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n}}$$)$$\mathrm{N}$$\mathrm{S}$$\hat{T}_{\alpha}$$Y_{\lambda}:= \frac{\mathrm{s}_{\alpha\delta}}{\mathrm{s}_{1\delta}}\sqrt{\frac{D_{\lambda}}{\mathrm{D}_{\alpha}}}$$\lambda$$Y_{\lambda} \mathrm{Y}_{\mu}=\sum_{\nu}M_{\lambda\mu}\nu \mathrm{Y}_{\nu}$$\delta$$d$$\frac{\psi_{d}^{(\lambda)}}{\psi_{d}^{(1)}}$$\frac{\psi_{d}^{(\lambda)}}{\psi_{d}^{(1)}}=\frac{\mathrm{S}_{\alpha\delta}}{\mathrm{S}_{1\delta}}\sqrt{\frac{D_{\lambda}}{\mathrm{D}_{\alpha}}}$

.

(5.5)

In particular, the Krein parameter of the Pasquier algebra reads$\hat{k}_{\lambda}=D_{\lambda}\mathrm{D}_{\alpha}$

to be compared with the formula,$\hat{k}_{\lambda}=D_{\lambda}^{2}$

if

(5.6)

$\lambda\in\hat{T}_{\alpha}$of sect. 3.1, example 2, valid for the fusion

algebras, i.e. for the cases for which the blocks
contain only one exponent. Let
denote the subset of vertices for which (5.5) holds. Each of them may be identified witha weight of the extended algebra. Further analysis [19] reveals that:
$A$

$T$

$T_{\alpha}$$d$$\delta$1)$\forall d\in T,$$\delta$

the corresponding extended weight, and for$\frac{\psi_{d}^{(\lambda)}}{\psi_{1}^{(\lambda)}}=\frac{\mathrm{S}_{\delta\alpha}}{\mathrm{S}_{1\alpha}}$and$\lambda\in\hat{T}_{\alpha}$$\psi_{1}^{(\lambda)}=S_{1\lambda}\mathrm{S}_{1\alpha}$2) one is precisely in the conditions of sect. 3.2: the set

one has;defines a(5.7)-subalgebra of the$N$ algebra. In
the cases of (5.2) discussed here, the $M$ and $N$ structure constants are nonnegative (see Fact 3 of sect. 1). One may apply the theorem of Bannai and Ito: the dual
subalgebra is associated with a special set which is the block of the identity representation$\hat{T}$$T$$\mathrm{C}$105

and it defines a partition of the set of exponents into classes . Finally equation (5.4) may
be seen to be equivalent to equation (3.9) (or rather its dual), if one takes into account thestructure constants and the explicit
andchange of normalization between theexpressions of the Krein parameters (5.6).
Thus behind the modular invariants (5.4), there is again a structure of C-algebrasand subalgebras. This had been first pointed out in [20], and then the more systematic$T_{\alpha}$$M_{\lambda\mu}\nu$$q_{\lambda\mu}\nu$

discussion of [19] has shown that this follows from the basic equation (5.4), and thatit yields a way to determine the expressions of some eigenvectors from conformal data
(quantum dimensions).

6. Conclusion and perspectives

The purpose of this lecture was to present the concept of -algebra and to illustrate its
utility in two contexts: the discussion of reflection groups and their foldings on the one
hand, and the structure of conformal field theories, on the other.
Note that these two seemingly disparate problems are in fact related in the framework$\mathrm{C}$of 2-dimensional topological field theories. For those theories, or at least for those that are
obtained by twisting a $N=2$ superconformal coset field theory, one has two approaches at
one’s disposal: the discussion of the (super)conformal field theory following lines analogous
to the discussion of sect. 5; and the analysis of the Witten-Dijkgraaf-Verlinde-Verlinde
equations [1], for which Dubrovin [16] has shown the appearance of monodromy groups
generated by reflections. In fact the concept of -algebra seems to be underlying in a$\mathrm{C}$natural way the whole discussion of topological field theories.
Note also that in the two discussions of the previous sections, the same C-algebras(based on the Pasquier algebra of the Dynkin diagrams) have been used in two different
ways: in one case (folding), we have been looking at the -subalgebras of the $M$ algebra(subject to some constraint); in the other (rcft), it is rather some subalgebra of the $N$.
algebra that has determined the special set of vertices, and by duality the blocks$\mathrm{C}$$T$$T_{\alpha}$

One issue that requires clarification is the role of positivity. We have from the start
restricted our attention to the subcases of the $ADE$ list that have certain positivity prop(see sect. 2). The main benefit has been the possibility to use the theorem of Bannai
and Ito (sect. 3.2). It is possible to relax the positivity assumption in the discussion offolding of graphs and groups: what is really crucial is eq. (3.8), see [18]. In the case of$\mathrm{e}\mathrm{r}\mathrm{t}\dot{\mathrm{i}}\mathrm{e}\mathrm{s}$rcft, it is less clear how to proceed and what replaces (5.4). In that case, however, we106know that any theory with a non block diagonal modular invariant (e.g. $(2.2b)$ ) may beobtained from a block diagonal one ( $(2.2a)$ in that case) by an automorphism of the fusion
algebra [21]. The proper incorporation of that fact in the present considerations remains
to be done.
As already mentionned, the very good news is that all this discussion is not limited to
the $sl(2)$ -ADE cases to which I have restricted myself here for simplicity. On the contrary,both the folding of generalized Dynkin diagrams associated with $sl(N)$ and the block
structure of
RCFT may be discussed in quite general terms. The -algebra method
enables one to find in a fairly systematic way the possible foldings of these generalized
diagrams that respect some general properties, and in the second context, it gives nontrivial relations between conformal data (fusion coefficients and quantum dimensions) andeigenvectors of the adjacency matrices. It may even enable one to construct the graphfrom these data. See [18] for the former subject and [19] for the latter.
$sl(N)\wedge$$\mathrm{C}$Acknowledgements It is a pleasure to thank the organisers of this symposium for avery interesting and profitable meeting. I have benefited from an interesting conversation
with T. Yano. Stimulating discussions with M. Bauer are also gratefully acknowledged.
Finally I want to recall that most of the results presented here have been worked out incollaborations with P. Di Francesco and V. Petkova.

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