Quantum Groups and Fuss-catalan Algebras

The categories of representations of compact quantum groups of automorphisms of certain inclusions of finite dimensional C∗-algebras are shown to be isomorphic to the categories of Fuss-Catalan diagrams. A Fuss-Catalan diagram is a planar diagram formed by an upper row of 4m points, a lower row of 4n points and by 2m+2n non-crossing strings joining them. Both rows of points are colored from left to right in the following standard way white, black, black, white, white, black, black,... and the strings have to join pairs of points having the same color. The tensor C-category FC is defined as follows. The objects are the positive integers. The arrows between m and n are the linear combinations of such diagrams. The operations ◦, ⊗ and ∗ are induced by vertical and horizontal concatenation and upside-down turning of diagrams. With the following rule: erasing a black/white circle is the same as multiplying by β/ω, where β and ω are some fixed positive numbers. In [6] Bisch and Jones study the algebras FC(m,m) are prove that they are isomorphic to the algebras associated to intermediate subfactors of indices β and ω. Let B ⊂ D be an inclusion of finite dimensional C-algebras and let φ be a state on D. Following Wang ([12]) one can construct a quantum group Gaut(B ⊂ D,φ): the biggest compact quantum group acting on D such that B and φ are left invariant. In this paper we prove that, under suitable assumptions on φ, the category of representations of Gaut(B ⊂ D,φ) is isomorphic to the completion of FC. The particular case B = C and φ = trace is studied in [3]. The paper is organised as follows. In §1, §2 and §3 we recall some constructions relating planar diagrams and representations. In §4, §5 and §6 we prove the main result. In §7 we discuss the relation with subfactors. 1. The Jones projection A N-algebra is by definition a tensor C-category having (N,+) as monoid of objects. If C is a N-algebra we use the notations C(m,n) = Hom(m,n) C(m) = End(m) Associated to any object O in a tensor C-category is the N-algebra NO given by NO(m,n) = Hom(O, O) If C is a N-algebra, its diagonal ∆C is defined by ∆C(m) = C(m) and ∆C(m,n) = ∅ if m 6= n. It is a N-algebra. A N-algebra C is said to be diagonal if C = ∆C. 1