A "well-known" fact is that a ${C}_q[U(1)]$-coaction on a vector space $V$ induces a $Z$-grading. I don't see why this is so. Clearly, a ${C}_q[U(1)]$-grading will induce a $Z$-grading. But why does a ${C}_q[U(1)]$-coaction induce a ${C}_q[U(1)]$-grading? In other words, Why should every $v \in V$ be equal to a sum $\sum_i v_i$, for which $\Delta (v_i) = v_i \otimes k^n$?

Moreover, what happens when we "move up" to a ${C}_q[U(2)]$-coaction, or a ${C}_q[U(N)]$-coactions? Does this induce a ${C}_q[U(2)]$-grading, and a ${C}_q[U(N)]$-grading?