论文标题
DG代数结构在量子仿射$ n $ -space $ \ mathcal {o} _ { - 1}（k^n）$
DG Algebra structures on the quantum affine $n$-space $\mathcal{O}_{-1}(k^n)$
论文作者
论文摘要
令$ \ Mathcal {a} $为连接的Cochain DG代数，其基础分级代数$ \ Mathcal {a}^{\＃} $是量子$ n $ n $ -space $ \ Mathcal $ \ Mathcal {O}我们计算$ \ mathcal {a} $的所有可能的微分结构，并表明存在$$ \ {\ text {Cochain DG之间存在一对一的对应关系Algebra} \，\，\ Mathcal {a} \，| \，\ Mathcal {a}^{\＃} = \ Mathcal {O} _ { - 1}（k^n）\} $ n \ times n \ times n \ times n \ times n \ times n $ matrices $ m_n（k）$ m_n（k）$。对于任何$ m \在m_n（k）$中，我们编写$ \ Mathcal {a} _ {\ Mathcal {o} _ { - 1}（k^3）}（m）$ for DG代数对应于它。我们还研究了这些非共同DG代数的同构问题。对于$ n \ le 3 $的情况，我们检查了它们的同源性能。与$ n = 2 $的情况不同，我们发现并非所有当时都是$ n = 3 $的calabi-yau。尽管如此，我们还是认识到那些calabi-yau的情况。简而言之，我们解决了如何判断给定的DG代数$ \ Mathcal {a} _ {\ Mathcal {o} _ { - 1}（k^3）}（m）$的问题。
Let $\mathcal{A}$ be a connected cochain DG algebra, whose underlying graded algebra $\mathcal{A}^{\#}$ is the quantum affine $n$-space $\mathcal{O}_{-1}(k^n)$. We compute all possible differential structures of $\mathcal{A}$ and show that there exists a one-to-one correspondence between $$\{\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.
