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

TL;DR

This paper classifies all differential structures on the quantum affine n-space as DG algebras, explores their isomorphisms, and investigates their homological properties, including Calabi-Yau conditions for low dimensions.

Contribution

It provides a complete classification of DG algebra structures on quantum affine space and analyzes their homological and Calabi-Yau properties.

Findings

One-to-one correspondence between DG structures and n×n matrices.

Not all DG algebras for n=3 are Calabi-Yau, unlike for n=2.

Criteria established for determining Calabi-Yau property.

Abstract

Let $\mathcal{A}$ be a connected cochain DG algebra, whose underlying graded algebra $\mathcal{A}^{\#}$ 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 $$\{\text{cochain DG algebra}\,\,\mathcal{A}\,|\,\mathcal{A}^{\#}=\mathcal{O}_{-1}(k^n)\}$$ 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 to it. We also study the isomorphism problems of these non-commutative DG algebras. For the cases $n\le 3$, 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 $\mathcal{A}_{\mathcal{O}_{-1}(k^3)}(M)$ is Calabi-Yau.