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arXiv 2011.11107math.RT
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对偶扩张代数上标准模的$\operatorname{Ext}$ -代数

The $\operatorname{Ext}$-algebra of standard modules over dual extension algebras

Markus Thuresson

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35 pages
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摘要：

给出了两个有向代数$B$和$A$的对偶扩张代数$Λ$上的标准模的$\operatorname{Ext}$ -代数$\operatorname{Ext}_Λ^\ast(Δ,Δ)$与$\operatorname{Ext}$ -代数$\operatorname{Ext}_B^\ast(\mathbb{L},\mathbb{L})$和$A$的对偶扩张代数之间的结合代数的同构.在这些$\operatorname{Ext}$ -代数上存在自然的$A_\infty$ -结构，并且在对$B$的某些技术假设下，我们完全用$\operatorname{Ext}_B^\ast(\mathbb{L},\mathbb{L})$上的结构来描述$\operatorname{Ext}_Λ^\ast(Δ,Δ)$上的结构.作为一个例子，我们计算这些$A_\infty$ -结构明确的情况下，$B=A=K\mathbb{A}_n /(\operatorname{rad}K\mathbb{A}_n)^\ell$。

英文摘要：

We exhibit an isomorphism of associative algebras between the $\operatorname{Ext}$-algebra $\operatorname{Ext}_Λ^\ast(Δ,Δ)$ of standard modules over the dual extension algebra $Λ$ of two directed algebras $B$ and $A$ and the dual extension algebra of the $\operatorname{Ext}$-algebra $\operatorname{Ext}_B^\ast(\mathbb{L},\mathbb{L})$ with $A$. There are natural $A_\infty$-structures on these $\operatorname{Ext}$-algebras, and, under certain technical assumptions on $B$, we describe that on $\operatorname{Ext}_Λ^\ast(Δ,Δ)$ completely in terms of that on $\operatorname{Ext}_B^\ast(\mathbb{L},\mathbb{L})$. As an example, we compute these $A_\infty$-structures explicitly in the case where $B=A=K\mathbb{A}_n /(\operatorname{rad}K\mathbb{A}_n)^\ell$.

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