Leavitt path algebras, $B_\infty$-algebras and Keller's conjecture for singular Hochschild cohomology

Abstract

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 $\Lambda$. Combining this isomorphism with a description of the dg singularity category of $\Lambda$ in terms of the dg perfect derived category of $L$, we verify Keller's conjecture for the singular Hochschild cohomology of $\Lambda$. More precisely, we prove that there is an isomorphism in $\mathrm{Ho}(B_\infty)$ between the singular Hochschild cochain complex of $\Lambda$ and the Hochschild cochain complex of the dg singularity category of $\Lambda$. 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 $\Lambda$ by one-point (co)extensions and singular equivalences with levels. These algebras include all finite dimensional gentle algebras.