Leavitt路代数、$B_\infty$-代数与奇异Hochschild上同调的Keller猜想
Leavitt path algebras, $B_\infty$-algebras and Keller's conjecture for singular Hochschild cohomology
AI总结：
针对无汇点有限箭图，本文通过建立$B_\infty$-代数同伦范畴中的复形同构，验证了根平方零代数的Keller猜想，证明其在单点（余）扩张等变换下不变，结论覆盖所有有限维gentle代数。
摘要：
针对无汇点的有限箭图，我们在$B_\infty$-代数的同伦范畴$\mathrm{Ho}(B_\infty)$中，建立了Leavitt路代数$L$的Hochschild上链复形与相应根平方零代数$\Lambda$的奇异Hochschild上链复形之间的同构。将该同构与借助$L$的dg完美导出范畴对$\Lambda$的dg奇点范畴的刻画相结合，我们验证了关于$\Lambda$奇异Hochschild上同调的Keller猜想。更准确地说，我们证明了在$\mathrm{Ho}(B_\infty)$中，$\Lambda$的奇异Hochschild上链复形与$\Lambda$的dg奇点范畴的Hochschild上链复形之间存在同构。证明的一个组成部分是如下关于$B_\infty$-代数的对偶定理：对任意$B_\infty$-代数，其相反$B_\infty$-代数与其转置$B_\infty$-代数之间存在自然的$B_\infty$-同构。我们证明了Keller猜想在单点（余）扩张和带层次的奇异等价下具有不变性，因此Keller猜想对所有由$\Lambda$通过单点（余）扩张和带层次的奇异等价归纳得到的代数成立，这类代数包含所有有限维gentle代数。
英文摘要：
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.
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