Translation Quiver in Representation Theory

• Translation quiver is defined as a directed graph equipped with a translation operation (τ) that organizes indecomposable modules via mesh relations and Auslander–Reiten sequences.
• It provides a combinatorial framework to study module categories by depicting indecomposable modules and irreducible morphisms in a clear, structured manner.
• Extensions such as n-translation quivers and related geometric models illustrate its role in higher representation theory, tilting mutations, and quiver variety constructions.

A translation quiver is a directed graph equipped with a translation operation (typically denoted by τ) that organizes its vertex set—often representing isomorphism classes of indecomposable modules—by relating almost split sequences, morphisms, or mesh relations in the context of the representation theory of finite-dimensional algebras. The translation function encapsulates the Auslander–Reiten translation, which connects the internal combinatorics of representations with homological and categorical properties of the module category. Translation quivers arise pervasively in the paper of Auslander–Reiten theory, mesh categories, higher representation theory, and geometric models such as translation quiver varieties.

1. Formal Definition and Structure

Let $Q = (Q_0, Q_1)$ be a quiver, where $Q_0$ is the set of vertices and $Q_1$ the set of arrows. A translation quiver is the data $(Q, \tau)$, where $\tau: Q_0 \setminus P \to Q_0 \setminus I$, for subsets $P$ (projective vertices) and $I$ (injective vertices), satisfying:

• For each $i\in Q_0 \setminus P$, there exists a unique (up to scalar) path of length $n+1$ from $\tau(i)$ to $i$, and any two such are linearly dependent (for $n=1$, this recovers the classical case).
• Mesh relations: For vertices not in $P \cup I$, every irreducible morphism entering $i$ is part of a mesh—i.e., fits into a commutative diamond encoding almost split sequences.

This structure generalizes to $n$-translation quivers, defined via a translation $T$ and prescribed combinatorics of maximal bound paths of length $n+1$. The mesh (or diamond) relations are critical for encoding the Auslander–Reiten sequences among modules.

2. Roles in Representation Theory

The translation quiver (or Auslander–Reiten quiver) encodes:

• Indecomposable modules as vertices, irreducible morphisms as arrows.
• The translation $\tau$ matches almost split (AR) sequences: For non-projective indecomposables $M$, the socle of the projective cover of $M$ is isomorphic to $\tau(M)$.
• Families of indecomposables: postprojective, preinjective, regular (as in hereditary algebras of type $\tilde{A}_n$ (Bruestle et al., 2013)).
• Mesh relations describe how morphisms "compose" in almost split sequences and correspond to relations in the mesh category.

In higher representation theory, $n$-translation quivers and $n$-translation algebras encode higher almost split sequences (or $n$-AR sequences), generalizing these combinatorics to higher dimensions (Guo, 2014).

3. Construction Techniques and Examples

The classical construction is $\mathbb Z Q$ for a quiver $Q$, whose vertices are $(i, k)$ with $i\in Q_0$, $k \in \mathbb Z$, with arrows repeating those of $Q$ at each level $k$, and the translation $(i, k)\mapsto (i, k-1)$. Extensions include:

• For hereditary algebras of type $\tilde{A}_n$: the infinite translation quiver $\widetilde{K}$ encodes the module category as walks along a cyclic quiver; mesh and reflection functor relations organize the regular, preinjective, and postprojective blocks (Bruestle et al., 2013).
• Higher translation quivers: Trivial extensions and smash products yield $(n+1)$-translation quivers from $n$-translation ones, adding "returning arrows" $i\to T(i)$ and mesh relations with length $n+1$, connecting with $(n-1)$-almost split sequences in the quadratic dual (Guo, 2014).

Examples include the translation quivers arising from extended Dynkin diagrams (affine type), the mesh categories attached to these, and stable translation quivers built using infinite (repetitive) covers.

4. Deeper Algebraic and Geometric Connections

Translation quivers underpin numerous constructions:

• The mesh algebra (or preprojective algebra) of a translation quiver is $k Q / (\mathrm{mesh}\ \mathrm{relations})$, encoding the AR-sequences as relations (Mozgovoy, 2019).
• Moduli spaces (quiver varieties) such as Nakajima quiver varieties and their generalizations (translation quiver varieties) use translation quiver/combinatorics to organize their geometry (fixed points under toric actions are again translation quiver varieties) (Mozgovoy, 2019).
• Coverings and embeddings: Regular quiver coverings and embeddings (for instance, of McKay quivers) induce translation quiver structures; for example, embedding a finite subgroup $G\subset SL(m,\mathbb C)$ into $GL(m,\mathbb C)$ gives a regular covering which aligns with translation quiver topology (Guo, 2010).

5. Interplay with Mutation, Tilting, and Categorification

Translation quivers formalize operations essential in higher representation theory and categorification:

• $n$-APR tilting corresponds to "mutating" a sink (or source) in a translation quiver, replacing the vertex with its $n$-translation image, and producing derived equivalence via the T-mutation; the new quiver encodes the tilted (or cotilted) algebra (Guo et al., 2019, Guo et al., 2017).
• In cluster theory, quiver mutations and mutation loops can be seen as "translation-like" operations; translation quivers formalize this structure, and their associated partition $q$-series reproduce character formulas in affine Lie algebras (Kato et al., 2014).
• In higher-dimensional examples, $\tau$-slice algebras are defined as endomorphism algebras of slices through a translation quiver, with recursive mutations yielding higher quasi-Fano algebras (Guo et al., 2017).

6. Applications and Extension to Abelian and Geometric Categories

Translation quivers and their generalizations appear in contexts beyond module categories:

• In twisted quiver representation theory, one replaces vector spaces at vertices with objects in abelian categories and arrows with functors; the "twisting" acts as a categorical translation, and standard resolutions/coresolutions mirror AR theory (Mozgovoy, 2018).
• In geometric representation theory, translation quivers appear as the foundation for the structure of generalized quiver Hecke algebras, with the combinatorics of the quiver's "translation" reflected in the grading and relations among the algebra's generators (Sauter, 2013).

7. Summary Table: Translation Quiver Data

Structure	Role	Example/Formula
$(Q, \tau)$	Translation quiver	$\tau(i)$: AR translation, mesh relations among vertices
$n$-translation quiver	Higher AR theory, $n$-sequences	For each $i$, unique path of length $n+1$ from $T(i)$ to $i$
Mesh algebra	Encodes AR sequences	$kQ / (\text{mesh relations})$
T-mutation	Tilting mutation	Replace $i$ with $T^{-1}(i)$ (sink) or $T(i)$ (source)
Quiver variety	Geometric realization	Moduli of $\Pi(\Gamma)$-modules with mesh relation $r=\sum \varepsilon(a) a \sigma(a)$

Translation quivers thus serve as the central index set and combinatorial skeleton for the organization of indecomposable objects, morphisms, and homological invariants in representation theory and related geometric and combinatorial settings. Their structure and mutations encode the passage between derived or module categories, realization of categorical symmetries, and underlie the construction of moduli spaces, categorifications, and representation-theoretic correspondences.