# Quantum (group) version of ${\mathbb Z}^n$?

As we know there are quantum analogue of tori called quantum tori generated by noncommuting operators $$(A_1,\dots,A_n)$$ with $$A _iA_j=A_jA_ie^{2\pi i\alpha}$$ where $$\alpha$$ is a irrational number as a universal $$C^*$$ algebra and a similar definition of a quantum Euclidean space, can we define a notion of quantum integer space, which completes the circle of classical groups? Also can we have a notion of quantum $$p$$-adic group? Also can we define some sort of Fourier transform on these spaces, as we can do for quantum tori and quantum Euclidean spaces?

I'm not sure what you mean by "quantum integer space", but if you're asking for an analog of the Fourier correspondence between $$\mathbb{T}^2$$ and $$\mathbb{Z}^2$$ I think the answer is no. We do have a notion of duality for quantum groups, but there is no reasonable way to put a quantum group structure on the quantum torus.