Fusion Rules for Representations of Compact Quantum Groups

The compact quantum groups are objects which generalise at the same time the compact groups, the duals of discrete groups and the q−deformations (with q > 0) of classical compact Lie groups. A compact quantum group is an abstract object which may be described by (is by definition the dual of) the algebra of “continuous functions on it”, which is a Hopf C-algebra. A system of axioms for Hopf C-algebras which leads to a satisfactory theory of compact quantum groups (e.g. a theorem stating the existence of the Haar measure) was found by Woronowicz at the end of the 80’s. The representation theory of compact quantum groups gives rise to rich combinatorial structures. By Woronowicz’s analogue of the Peter-Weyl theory, each (finite dimensional unitary) representation of a compact quantum group G is completely reducible. In particular given two irreducible representations a and b, their tensor product decomposes in a unique way (up to equivalence) as a sum of irreducible representations a⊗ b ≃ c+ d+ e+ · · ·