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Let $\mathcal{A}$ be a connected cochain DG algebra, whose underlying gradedalgebra $\mathcal{A}^{\#}$ is the quantum affine $n$-space$\mathcal{O}_{-1}(k^n)$. We compute all possible differential structures of$\mathcal{A}$ and show that there exists a one-to-one correspondence between$$\{\text{cochain DGalgebra}\,\,\mathcal{A}\,|\,\mathcal{A}^{\#}=\mathcal{O}_{-1}(k^n)\}$$ and the$n\times n$ matrices $M_n(k)$. For any $M\in M_n(k)$, we write$\mathcal{A}_{\mathcal{O}_{-1}(k^3)}(M)$ for the DG algebra corresponding toit. We also study the isomorphism problems of these non-commutative DGalgebras. For the cases $n\le 3$, we check their homological properties. Unlikethe case of $n=2$, we discover that not all of them are Calabi-Yau when $n=3$.In spite of this, we recognize those Calabi-Yau ones case by case. In brief, wesolve the problem on how to judge whether a given such DG algebra$\mathcal{A}_{\mathcal{O}_{-1}(k^3)}(M)$ is Calabi-Yau.

DG Algebra structures on the quantum affine $n$-space $\mathcal{O}_{-1}(k^n)$