arXiv subjects

Maoyun Zhang

Publications and source records attributed to Maoyun Zhang.

3 recordsLinked to original sources

arXiv2021

Homological properties of $3$-dimensional DG Sklyanin algebras

In this paper, we introduce the notion of DG Sklyanin algebras, which are connected cochain DG algebras whose underlying graded algebras are Sklyanin algebras. Let $\mathcal{A}$ be a $3$-dimensional DG Sklyanin algebra with $\mathcal{A}^{\#}=S_{a,b,c}$, where $(a,b,c)\in \Bbb{P}_k^2-\mathfrak{D}$ and $$\mathfrak{D}=\{(1,0,0), (0,1,0),(0,0,1)\}\sqcup\{(a,b,c)|a^3=b^3=c^3\}.$$ We systematically study its differential structures and various homological properties. Especially, we figure out the conditions for $\mathcal{A}$ to be Calabi-Yau, Koszul, Gorenstein and homologically smooth, respectively.

math.RA↗

arXiv2021

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

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.

math.RA↗

arXiv2019

Some dimensions of DG polynomial algebras

Assume that $\mathcal{A}$ is a cochain DG polynomial algebra such that its underlying graded algebra $\mathcal{A}^{#}$ is a polynomial algebra generated by $n$ degree $1$ elements. We determine the DG Krull dimension, the global dimension, the ghost dimension and the Rouquier dimension of $\mathcal{A}$.

math.RA↗