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

Xiao-Wu Chen, Huanhuan Li, Zhengfang Wang

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 $Λ$. 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.

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 of -algebras between the Hochschild cochain complex of the Leavitt path algebra and the singular Hochschild cochain complex of the corresponding radical square zero algebra . Combining this isomorphism with a description of the dg singularity category of in terms of the dg perfect derived category of , we verify Keller's conjecture for the singular Hochschild cohomology of . More precisely, we prove that there is an isomorphism in between the singular Hochschild cochain complex of and the Hochschild cochain complex of the dg singularity category of . One ingredient of the proof is the following duality theorem on -algebras: for any -algebra, there is a natural -isomorphism between its opposite -algebra and its transpose -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 by one-point (co)extensions and singular equivalences with levels. These algebras include all finite dimensional gentle algebras.

Paper Structure

This paper contains 43 sections, 66 theorems, 404 equations, 9 figures.

Table of Contents

  1. Introduction
  2. The background
  3. The main results
  4. The structure of the paper
  5. DG categories and dg quotients
  6. DG categories and dg functors
  7. One-point (co)extensions and singular equivalences with levels
  8. The dg singularity category and acyclic complexes
  9. Quivers and Leavitt path algebras
  10. An introduction to $B_\infty$-algebras
  11. $A_\infty$-algebras and morphisms
  12. $B_\infty$-algebras and morphisms
  13. A duality theorem on $B_\infty$-algebras
  14. Brace $B_\infty$-algebras
  15. Gerstenhaber algebras
  16. ...and 28 more sections

Key Result

Theorem 1.1

Let $(A, m_n; \mu_{p, q})$ be a $B_\infty$-algebra. Then there is a natural $B_\infty$-isomorphism between the opposite $B_\infty$-algebra $A^{\rm opp}$ and the transpose $B_\infty$-algebra $A^{\rm tr}$.

Figures (9)

  • Figure 1: Three of the summands in $\Theta_6$
  • Figure 2: The $A_{\infty}$-product $m_k$ is on the left and the $A_{\infty}$-quasi-isomorphism $\iota_k$ is on the right, where the sums are taken over $BPT(k)$, the set of all planar rooted binary trees with $k$ leaves.
  • Figure 3: The $A_{\infty}$-product $m_k$ and $A_{\infty}$-quasi-isomorphism $\iota_k$.
  • Figure 4: The tree-like and cactus-like presentations of $f\in \overline{C}^{m-p}(\Lambda, \Omega_{\mathrm{nc}, R}^p(\Lambda))$.
  • Figure 5: The colimit maps $\theta_{*, R}$, where the straight line represents the identity map of $s \overline \Lambda$.
  • ...and 4 more figures

Theorems & Definitions (126)

  • Theorem 1.1: = Theorem \ref{['thm:dualityB']}
  • Theorem 1.2: = Theorem \ref{['thm:Keller-redu']}
  • Theorem 1.3: = Theorem \ref{['thm-main']}
  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Example 2.7
  • ...and 116 more