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