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The second Hochschild cohomology and deformations of Brauer graph algebras

Yuming Liu, Zhengfang Wang, Bohan Xing

TL;DR

This work delivers an explicit, basis-driven description of the second Hochschild cohomology $\mathrm{HH}^2(B_\Gamma)$ for bipartite Brauer graph algebras, organized into four standard cocycle types (A,B,C,D). It constructs a natural reduction system to compute $2$-cochains and provides precise dimension formulas tying $\mathrm{HH}^2$ to combinatorial and surface-data, including a geometric interpretation of corresponding deformations on surface models. The paper proves that each standard $2$-cocycle induces a formal deformation and clarifies when multiple deformations can be lifted simultaneously, with particular emphasis on the semisimple deformation of type (A) and line-field/boundary deformations of types (C) and (D). It further translates these algebraic deformations into surface-model operations, showing how type (A) yields semisimple matrix-block decompositions, type (C) aligns with line-field deformations, and type (D) corresponds to boundary compactifications, while exposing obstructions and orbifold-like phenomena in the non-bipartite setting. Together, these results bridge Hochschild cohomology, quiver reductions, and geometric surface models, enriching deformation theory for Brauer graph algebras and related $A_\infty$-categories.

Abstract

In this paper, we give an explicit description about the second Hochschild cohomology groups of bipartite Brauer graph algebras with trivial grading. Based on this, we provide geometric interpretations of deformations associated to some standard cocycles in terms of the surface models of Brauer graph algebras.

The second Hochschild cohomology and deformations of Brauer graph algebras

TL;DR

This work delivers an explicit, basis-driven description of the second Hochschild cohomology for bipartite Brauer graph algebras, organized into four standard cocycle types (A,B,C,D). It constructs a natural reduction system to compute -cochains and provides precise dimension formulas tying to combinatorial and surface-data, including a geometric interpretation of corresponding deformations on surface models. The paper proves that each standard -cocycle induces a formal deformation and clarifies when multiple deformations can be lifted simultaneously, with particular emphasis on the semisimple deformation of type (A) and line-field/boundary deformations of types (C) and (D). It further translates these algebraic deformations into surface-model operations, showing how type (A) yields semisimple matrix-block decompositions, type (C) aligns with line-field deformations, and type (D) corresponds to boundary compactifications, while exposing obstructions and orbifold-like phenomena in the non-bipartite setting. Together, these results bridge Hochschild cohomology, quiver reductions, and geometric surface models, enriching deformation theory for Brauer graph algebras and related -categories.

Abstract

In this paper, we give an explicit description about the second Hochschild cohomology groups of bipartite Brauer graph algebras with trivial grading. Based on this, we provide geometric interpretations of deformations associated to some standard cocycles in terms of the surface models of Brauer graph algebras.
Paper Structure (23 sections, 10 theorems, 77 equations, 8 figures)

This paper contains 23 sections, 10 theorems, 77 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Brauer graph algebras and surface models
  3. Review on Brauer graph algebras
  4. Brauer graphs
  5. Brauer graph algebras
  6. Brauer graph $A_\infty$-categories and surface models
  7. The second Hochschild cohomology groups of bipartite Brauer graph algebras
  8. Reduction systems of Brauer graph algebras
  9. General forms of $2$-cochains
  10. Let $(s,\varphi_s)$ be an element of type (a) in $R_\Gamma$
  11. Let $(s,\varphi_s)$ be an element of type (b) in $R_\Gamma$
  12. Let $(s,\varphi_s)$ be an element of type (c) in $R_\Gamma$
  13. Hochschild cohomology in degree $2$
  14. Deformations arising from the second Hochschild cohomology group
  15. Deformations of surface models of bipartite Brauer graph algebras
  16. ...and 8 more sections

Key Result

Theorem 1.1

$(\text{\rm see Corollary }dimhh2\text{ and Example }exam:localbrauer)$ Let $B_\Gamma=\mathbbm{k} Q_\Gamma/I_\Gamma$ be a bipartite Brauer graph algebra corresponding to the bipartite Brauer graph $(\Gamma,\mathbbm{m})$.

Figures (8)

  • Figure 3: An arc system on a disk with two punctures
  • Figure 7: Bipartite ribbon graph for each punctured surface with at least two punctures.
  • Figure 8: Boundaries and genera can provide generators in $\mathrm{H}^1(\Sigma)$.
  • Figure 9: The boundary in the bigon which can induce deformations of type (D).
  • Figure 10: Deformations induced by $\lambda'$ in surface model.
  • ...and 3 more figures

Theorems & Definitions (47)

  • Theorem 1.1
  • Proposition 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • ...and 37 more