DG algebra structures on the quantum affine n-space [formula omitted

Abstract

Let A be a connected cochain DG algebra, whose underlying graded algebra A# is the quantum affine n-space O−1(kn). We compute all possible differential structures of A and show that there exists a one-to-one correspondence between{cochain DG algebraA|A#=O−1(kn)} and the n×n matrices Mn(k). For any M∈Mn(k), we write AO−1(k3)(M) for the DG algebra corresponding to it. We also study the isomorphism problems of these non-commutative DG algebras. For the cases n≤3, we check their homological properties. Unlike the 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, we solve the problem on how to judge whether a given such DG algebra AO−1(k3)(M) is Calabi-Yau.