Non-commutative Euclidean structures in compact spaces

Based on results for real deformation parameter q we introduce a compact non-commutative structure covariant under the quantum group SOq(3) for q being a root of unity. To match the algebra of the q-deformed operators with necessary conjugation properties it is helpful to define a module over the algebra generated by the powers of q. In a representation where X-2 is diagonal we show how P-2 can be calculated. To manifest some typical properties, an example of a one-dimensional q-deformed Heisenberg algebra is also considered and compared with the non-compact case.