论文标题
Leavitt Path代数，$ b_ \ infty $ -Algebras和Keller对单数Hochschild的猜想
Leavitt path algebras, $B_\infty$-algebras and Keller's conjecture for singular Hochschild cohomology
论文作者
论文摘要
对于没有下水道的有限箭袋，我们在同型类别中建立同构类别$ \ mathrm {ho}（b_ \ infty）$ $ b _ {\ infty} $ - hochschild cochain cochain cophge path algebra $ l $ l $ l $ l $ l $ l $ l $ l $ l $ l $和平方gro n singular hochs cosere zer grouns grops zer groun和sige grops zer的代数。 $λ$。将这种同构与$ l $的DG Perfect派生类别的DG奇异性类别的描述相结合，我们验证了Keller对$λ$的Singular Hochschild共同体的猜想。更准确地说，我们证明$ \ mathrm {ho}（b_ \ infty）$ in $λ$的$ \ mathrm {ho}（b_ \ infty）$中存在同构，$λ$和$λ$的dg Singularity类别的Hochschild Cochain Cochain Complex之间存在同构。证明的一种成分是$ b_ \ infty $ -Algebras上的以下双重定理：对于任何$ b_ \ infty $ -Algebra，在其相对的$ b_ \ b_ \ infty $ -Algebra及其trapspose $ b_ \ b_ \ infty $ -Alfty $ -Alge-alge-alge-alge-alge-algebra之间都有天然的$ b_ \ infty $ - 异晶。 我们证明凯勒的猜想是在一分（CO）扩展和具有水平的奇异等价方面不变的。因此，凯勒（Keller）的猜想适用于那些通过$λ$从$λ$获得的代数（CO）扩展和具有水平的奇异等价的代数。这些代数包括所有有限的尺寸温和代数。
For a finite quiver without sinks, we establish an isomorphism in the homotopy category $\mathrm {Ho}(B_\infty)$ of $B_{\infty}$-algebras between the Hochschild cochain complex of the Leavitt path algebra $L$ and the singular Hochschild cochain complex of the corresponding radical square zero algebra $Λ$. Combining this isomorphism with a description of the dg singularity category of $Λ$ in terms of the dg perfect derived category of $L$, we verify Keller's conjecture for the singular Hochschild cohomology of $Λ$. More precisely, we prove that there is an isomorphism in $\mathrm{Ho}(B_\infty)$ between the singular Hochschild cochain complex of $Λ$ and the Hochschild cochain complex of the dg singularity category of $Λ$. One ingredient of the proof is the following duality theorem on $B_\infty$-algebras: for any $B_\infty$-algebra, there is a natural $B_\infty$-isomorphism between its opposite $B_\infty$-algebra and its transpose $B_\infty$-algebra. We prove that Keller's conjecture is invariant under one-point (co)extensions and singular equivalences with levels. Consequently, Keller's conjecture holds for those algebras obtained inductively from $Λ$ by one-point (co)extensions and singular equivalences with levels. These algebras include all finite dimensional gentle algebras.
