Bipartite Brauer Graph Algebras

• Bipartite Brauer graph algebras are finite-dimensional algebras defined from bipartite ribbon graphs with cyclic orderings and multiplicity functions.
• They exhibit rich homological behavior, with Hochschild cohomology (notably HH²) governing formal deformations and algebraic relations.
• These algebras bridge combinatorial, geometric, and topological methods, enabling studies of derived equivalences and surface deformation phenomena.

A bipartite Brauer graph algebra is a finite-dimensional algebra associated to a bipartite Brauer graph, itself a ribbon graph (graph with a prescribed cyclic order of edges at each vertex) equipped with a multiplicity function. These algebras admit a combinatorial and geometric formulation in terms of quivers with relations derived from the structure of the underlying graph, with connections to surface topology and deformation theory. Bipartite Brauer graph algebras admit powerful descriptions via their Hochschild cohomology, in particular $\mathrm{HH}^2$, which governs their formal deformations and is closely related to the geometry of ribbon surfaces into which the graph embeds (Liu et al., 11 Jan 2026).

1. Structure of Bipartite Brauer Graph Algebras

A ribbon graph $\Gamma = (V, H, s, \iota, \rho)$ consists of a finite vertex set $V$, a finite half-edge set $H$, an incidence map $s: H \to V$, a fixed-point-free involution $\iota: H \to H$ (whose orbits are the edges), and a cyclic permutation $\rho$ dictating the order of incident half-edges at each vertex. A Brauer graph is a ribbon graph equipped with a multiplicity function $m: V \to \mathbb{Z}_{>0}$. The graph is bipartite if the vertex set admits a 2-coloring $V = V_1 \sqcup V_2$ such that every edge joins a vertex in $V_1$ to one in $\Gamma = (V, H, s, \iota, \rho)$0.

Given $\Gamma = (V, H, s, \iota, \rho)$1, the corresponding Brauer graph algebra $\Gamma = (V, H, s, \iota, \rho)$2 is constructed as the quotient $\Gamma = (V, H, s, \iota, \rho)$3:

• The quiver $\Gamma = (V, H, s, \iota, \rho)$4 has vertex set $\Gamma = (V, H, s, \iota, \rho)$5 (edges of the graph), and arrow set $\Gamma = (V, H, s, \iota, \rho)$6, where $\Gamma = (V, H, s, \iota, \rho)$7. Each half-edge thus gives an arrow from its edge to the next edge around the same vertex.
• For each arrow $\Gamma = (V, H, s, \iota, \rho)$8, define a $\Gamma = (V, H, s, \iota, \rho)$9-cycle $V$0, where $V$1 and $V$2 is the valency at $V$3. Set $V$4.
• The ideal $V$5 is generated by two classes of relations:
  • (I) Zero-relations: $V$6 for arrows $V$7, $V$8 unless $V$9.
  • (II) Commutativity and power-relations: $H$0 for arrows with common terminal vertex; $H$1 for $H$2 ending at $H$3.

Because $H$4 is bipartite, these relations form a reduced Gröbner basis and the resulting reduction system $H$5 satisfies the Diamond Lemma for $H$6, ensuring uniqueness of reductions (Liu et al., 11 Jan 2026).

2. Trivial Grading and Homological Properties

Endowing $H$7 with the trivial grading—all arrows in degree 0—renders the algebra ungraded in the sense that all modules and derived structures inherit degree 0 and all homological computations (e.g., $H$8, Hochschild cohomology $H$9) coincide with their ungraded versions. In particular, the Chouhy–Solotar projective resolution of $s: H \to V$0 is homogeneous in degree 0, and the entire Hochschild cochain complex is concentrated in degree 0 (Liu et al., 11 Jan 2026).

3. The Second Hochschild Cohomology $s: H \to V$1

To compute $s: H \to V$2, fix the reduced Gröbner basis yielding the reduction system $s: H \to V$3 with tips $s: H \to V$4. The low-degree Hochschild cochain groups are

$s: H \to V$5

with explicit differentials $s: H \to V$6.

A basis for $s: H \to V$7 is described by standard cocycles, classified into four types:

Type	Description	Geometric Interpretation
(A)	Semisimple class: one cocycle $s: H \to V$8 for each edge (all agree)	Extending the line field to a vector field; yields semisimple algebra
(B)	Multiplicity classes: for each $s: H \to V$9 with $\iota: H \to H$0, $\iota: H \to H$1 cocycles	Insert powers in relations; control local multiplicity deformations
(C)	Cycle classes: one per edge not in a spanning tree, i.e., $\iota: H \to H$2	Twist the line field, changing boundary winding
(D)	Boundary classes: two per bigon in the surface realization with winding $\iota: H \to H$3	Compactify boundary components; smooth or orbifold caps

The precise cocycle representatives act by deforming commutativity, power, and boundary relations in $\iota: H \to H$4. The dimension formula is

$\iota: H \to H$5

where $\iota: H \to H$6 denotes the number of boundary components of winding $\iota: H \to H$7 in the ribbon surface $\iota: H \to H$8 associated to $\iota: H \to H$9 (Liu et al., 11 Jan 2026).

4. Deformations and Formal Moduli

Formal deformations of $\rho$0 are constructed by deforming the reduction system $\rho$1 via $\rho$2-cocycles $\rho$3 in

$\rho$4

This yields a flat formal $\rho$5-algebra whose fiber at $\rho$6 is $\rho$7, and all such deformations (up to isomorphism) are classified by $\rho$8.

Integration of the standard cocycles yields distinct families of deformations:

• Type (A): Deforms commutativity by $\rho$9, yielding a semisimple algebra $m: V \to \mathbb{Z}_{>0}$0.
• Type (B): Inserts powers $m: V \to \mathbb{Z}_{>0}$1 into commutativity relations for $m: V \to \mathbb{Z}_{>0}$2.
• Type (C): Flips signs in commutativity relations, corresponding to an $m: V \to \mathbb{Z}_{>0}$3-operation.
• Type (D): Compactifies boundaries in the ribbon surface; two cocycles correspond to smooth (disk) or orbifold (singular) caps.

These deformations commute pairwise and can be integrated in the reduction system up to arbitrarily high order in $m: V \to \mathbb{Z}_{>0}$4 (Liu et al., 11 Jan 2026).

5. Geometric and Categorical Interpretations

The minimal ribbon surface $m: V \to \mathbb{Z}_{>0}$5 into which $m: V \to \mathbb{Z}_{>0}$6 embeds, together with the induced line field $m: V \to \mathbb{Z}_{>0}$7, provides a geometric model for $m: V \to \mathbb{Z}_{>0}$8. The associated $m: V \to \mathbb{Z}_{>0}$9 Brauer graph category $V = V_1 \sqcup V_2$0 is derived equivalent to $V = V_1 \sqcup V_2$1.

• Type (A) deformation extends the line field to a nowhere-zero vector field, allowing boundaries to be capped and transforming the category into a product of matrix categories (Morita equivalence to semisimple algebras).
• Type (C) deformation corresponds to twisting the line field around a dual loop, i.e., changing the winding number of a boundary component.
• Type (D) deformation compactifies boundaries with winding $V = V_1 \sqcup V_2$2, yielding either a smooth disk (producing a smaller Brauer graph algebra up to Morita equivalence) or an orbifold disk (deformation toward gentle or skew-gentle algebras).

These geometric operations correspond precisely to the standard cocycles in $V = V_1 \sqcup V_2$3 and their effect on the algebraic and categorical structure (Liu et al., 11 Jan 2026).

6. Explicit Example

Consider a bipartite Brauer graph with two vertices $V = V_1 \sqcup V_2$4, $V = V_1 \sqcup V_2$5 connected by three parallel edges $V = V_1 \sqcup V_2$6 and multiplicities $V = V_1 \sqcup V_2$7. The quiver and relations are:

• $V = V_1 \sqcup V_2$8 (the three edges);
• Around $V = V_1 \sqcup V_2$9: $V_1$0; around $V_1$1: $V_1$2.
• $V_1$3 generated by $V_1$4, $V_1$5, $V_1$6 (indices modulo 3).

With reduction system $V_1$7,

$V_1$8

(If each vertex has $V_1$9, the count is adjusted accordingly for (B)). A basis of cocycles consists of one semisimple class (A), multiplicity-insertion classes (B) per vertex, and a cycle class (C) for the non-tree edge. Each cocycle integrates to a family of deformations in $\Gamma = (V, H, s, \iota, \rho)$00. The resulting deformations are visible at the level of the surface model (the annulus), corresponding to semisimple caps, multiplicity insertions, and cycle flips (Liu et al., 11 Jan 2026).

7. Significance and Applications

The explicit description of $\Gamma = (V, H, s, \iota, \rho)$01 and the classification of deformations in terms of standard cocycles establish precise algebraic and geometric relationships in the study of Brauer graph algebras. Formal deformations governed by these cocycles have direct correspondence to operations on the underlying ribbon surface—extending vector fields, twisting along cycles, and compactifying boundaries—which clarifies the interplay between noncommutative algebra, surface topology, and homological invariants.

The approach provides tools for the classification and mutation of derived equivalences, for explicit construction of families of gentle, skew-gentle, or semisimple algebras, and for understanding structural changes under categorical operations. The methods and explicit calculations are central to current research in the deformation theory of finite-dimensional algebras, the categorification of graph-theoretic data, and the geometric realization of algebraic structures in representation theory (Liu et al., 11 Jan 2026).