Leavitt path algebras, $B_\infty$-algebras and Keller's conjecture for singular Hochschild cohomology
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2020
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| author | Chen, Xiao-Wu Li, Huanhuan Wang, Zhengfang |
| author_facet | Chen, Xiao-Wu Li, Huanhuan Wang, Zhengfang |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2007_06895 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Leavitt path algebras, $B_\infty$-algebras and Keller's conjecture for singular Hochschild cohomology Chen, Xiao-Wu Li, Huanhuan Wang, Zhengfang Representation Theory Category Theory K-Theory and Homology Rings and Algebras 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. |
| title | Leavitt path algebras, $B_\infty$-algebras and Keller's conjecture for singular Hochschild cohomology |
| topic | Representation Theory Category Theory K-Theory and Homology Rings and Algebras |
| url | https://arxiv.org/abs/2007.06895 |