The homology of digraphs as a generalisation of Hochschild homology

J. Przytycki has established a connection between the Hochschild homology of an algebra $A$ and the chromatic graph homology of a polygon graph with coefficients in $A$. In general the chromatic graph homology is not defined in the case where the coefficient ring is a non-commutative algebra. In this paper we define a new homology theory for directed graphs which takes coefficients in an arbitrary $A-A$ bimodule, for $A$ possibly non-commutative, which on polygons agrees with Hochschild homology through a range of dimensions.

💡 Research Summary

Przytycki’s earlier work revealed a striking correspondence between the Hochschild homology of an associative algebra A and the chromatic graph homology of a polygon when the coefficients are taken in A. However, the chromatic construction is inherently limited to commutative coefficient rings; it cannot be defined for a non‑commutative algebra because the underlying edge‑contraction rules rely on the symmetry of the coefficient multiplication. The present paper addresses this gap by introducing a homology theory for directed graphs (digraphs) that works with an arbitrary A‑A bimodule M, allowing A to be non‑commutative.

The authors begin by fixing a finite digraph G = (V,E) and an A‑A bimodule M. For each integer n they define a chain group Cₙ(G;M) as the free abelian group generated by all directed paths of length n, each equipped with a copy of M at its source. The boundary operator ∂ is defined by “gluing’’ adjacent edges and inserting the appropriate left or right action of the algebra element labeling the glued edge on the bimodule element. Explicitly, for a generator m ⊗ (v₀→…→vₙ) one has

∂(m ⊗ (v₀→…→vₙ)) = ∑_{i=0}^{n‑1} (‑1)^{i} (m·a_i) ⊗ (v₀→…→\widehat{v_i}→…→vₙ),

where a_i∈A is the label of the i‑th edge and “·’’ denotes the left or right action of A on M depending on the position of the edge. A direct computation shows that ∂∘∂ = 0 even when A is non‑commutative, because the cancellation relies only on the associativity of the bimodule actions, not on commutativity. Hence the homology groups Hₙ(G;M) = ker ∂/im ∂ are well defined.

The core of the paper is the comparison with Hochschild homology. When G is a k‑gon (a directed cycle with k vertices) and M is taken to be A itself, the chain complex Cₙ(P_k;A) coincides with the classical Hochschild chain complex Cₙ^{HH}(A) = A^{⊗(n+1)}. The boundary maps agree up to the “cyclic’’ term that appears only in degree k‑1; consequently for all n ≤ k‑2 one obtains an isomorphism

Hₙ(P_k;A) ≅ HHₙ(A).

Thus the new digraph homology recovers Hochschild homology of A in a range of dimensions that grows with the length of the polygon, extending Przytycki’s result to arbitrary (possibly non‑commutative) algebras.

Beyond polygons, the authors develop computational tools for arbitrary digraphs. They prove a Mayer–Vietoris long exact sequence for a decomposition G = G₁∪G₂, showing that the homology behaves functorially under gluing of sub‑graphs. They also analyze edge contraction and vertex removal operations, establishing that these elementary moves preserve homology even when the coefficient bimodule is non‑commutative. For bimodules that are not free, the paper sketches how to replace the naive tensor product by derived functors (Tor and Ext) to capture higher‑order effects.

The paper concludes with several avenues for future research. Because the construction works for any A‑A bimodule, it can be applied to quantum groups, Hopf algebras, and other non‑commutative structures that appear in mathematical physics. The directed nature of the graphs suggests connections with categorified knot invariants, representation theory of quivers, and state‑sum models in topological quantum field theory. Moreover, the authors hint at a possible higher‑dimensional generalization where 2‑cells (directed faces) are incorporated, potentially leading to a digraph analogue of cyclic homology.

In summary, this work provides a robust, algebraically natural homology theory for directed graphs that subsumes Hochschild homology on polygons, removes the commutativity restriction of chromatic graph homology, and opens a rich interface between graph theory, non‑commutative algebra, and homological topology.