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
)
U(d)
as automorphisms of the Cuntz algebra
O
d
O_d
.
In particular we study the fixed point subalgebra under the coaction of the quantum compact groups
U
q
(
d
)
U_q(d)
on the Cuntz algebra
O
d
O_d
by extending to any dimension
d
>
∞
d>\infty
a result of Konishi (1992).
Furthermore we give a description of the fixed point subalgebra under the coaction of
S
U
q
(
d
)
SU_q(d)
on
O
d
O_d
in terms of generators.