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We study coactions of Hopf algebras coming from compact quantum groups on the Cuntz algebra. These coactions are the natural generalization to the coalgebra setting of the canonical representation of the unitary matrix group U(d) as automorphisms of the Cuntz algebra Od. In particular we study the fixed point subalgebra under the coaction of the quantum compact groups Uq(d) on the Cuntz algebra Od by extending to any dimension d < oc a result of Konishi (1992). Furthermore we give a description of the fixed point subalgebra under the coaction of SUq (d) on Od in terms of generators.

Coactions of Hopf algebras on Cuntz algebras and their fixed point algebras