Leavitt Path Algebras, 𝐵∞-Algebras, and Keller’s Conjecture for Ssingular Hochschild Ccohomology

For a finite quiver without sinks, we establish an isomorphism in the homotopy category H o (B ∞) Ho (B_) of B ∞ B -algebras between the Hochschild cochain complex of the Leavitt path algebra L 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 L, we verify Keller’s conjecture for the singular Hochschild cohomology of Λ. More precisely, we prove that there is an isomorphism in H o (B ∞) Ho (B_) 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 ∞ B_ -algebras: for any B ∞ B_ -algebra, there is a natural B ∞ B_ -isomorphism between its opposite B ∞ B_ -algebra and its transpose B ∞ B_ -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.