International Congress of Mathematicians

780 D. BischObserve that by construction planar algebras are closely related to invariantsfor graphs, knots and links and to the pictorial formalism commonly used in thetheory of integrable lattice models in statistical mechanics.4. Fuss-Catalan algebrasJones and I discovered in [7] a new hierarchy of finite dimensional algebras,which arise as the higher relative commutants of subfactors when intermediate subfactorsare present. These algebras have a number of interesting combinatorialproperties and they have recently been used to construct new integrable latticemodels and new solutions of the Yang-Baxter equation ([15], [29]).We show in [7] that a chain of fc — 1 intermediate subfactors N c Pi C-P 2 C ...Pk-i C M leads to a tower of algebras (FC n (ai,... ,ak)) n>Q , whichdepend on fc complex parameters ai,...,ak- The dimensions of these algebras aregiven by the generalized Catalan numbers or Fuss-Catalan numbers j^+ìi « ")and we therefore call these algebras the Fuss-Catalan algebras. If no intermediatesubfactor is present, i.e. F = N or Fj = M for all i, then one finds the wellknownTemperley-Lieb algebras (case fc = 1) [19]. The additional symmetry comingfrom the intermediate subfactor is captured completely by these new algebras andit is proved in [7] (see also [8]) that they constitute the minimal symmetry presentwhenever an intermediate subfactor occurs. See also [26].Let us explain in more detail what happens in the case of just one intermediatesubfactor. We consider N c F c M, an inclusion of Hi factors with finite Jonesindex, and construct the associated tower of of Hi factors as in section 2. One ob-Pi e\ P'2 62tains an inclusion of Hi factors N c P C M c Pi C Mi C F 2 c M 2 c ..., wherethe Pi's are the orthogonal projections from L 2 (Afj_i) onto L 2 (Fj_i) (P 0 = P,M 0 = M) and the intermediate subfactors F are the von Neumann algebras generatedby Mj_i and p,. The algebra IA n (a, ß) = Alg(l,ei,... ,e n _i,pi,... ,p n _i),generated by the e,'s and the p,'s, is a subalgebra of N' n M n _i. It can be shownto depend only on the two indices a = [P : N] and ß = [M : P], and not on theparticular position of F in N c M. The projections e, and pj satisfy again somerather nice commutation relations (see [7] for details). In order to describe thestructure of these algebras let us for the moment consider the complex vector spaceFC n (a, b), spanned by labelled, planar diagrams of the form2n marked pointsa b b a a b b ab a a b b awhere a, b £ C\{0} are fixed. There is a natural multiplication of these diagrams,which makes FC n (a, 6) into an associative algebra (see [25]). To obtain Di • D 2 put