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.

v3: 92 pages, 9 figures