Tame Hereditary Algebras

• Tame hereditary algebras are finite-dimensional hereditary algebras whose indecomposable modules appear in finitely many one-parameter families, reflecting controlled infinite representation type.
• They are Morita equivalent to path algebras of extended Dynkin quivers, decomposing into preprojective, regular (tube-based), and preinjective components.
• They underpin advanced studies in tilting theory, derived category recollements, and Hall algebras, linking representation theory to affine Lie algebras and combinatorial structures.

Tame hereditary algebras are finite-dimensional hereditary algebras over a field whose indecomposable modules, in every dimension $d$, occur in finitely many one-parameter families and a finite set outside these families—a representation-theoretic property reflecting a profound geometric and combinatorial structure. They are Morita equivalent to path algebras of extended Dynkin (Euclidean) quivers and form the central class of algebras of tame representation type in the Drozd–Dlab–Ringel dichotomy. The subject interweaves representation type, combinatorics of quivers, stratifications via tubes, tilting theory, derived category recollements, Hall algebra theory, and Coxeter lattice combinatorics.

1. Structural Definition and Classification

A $k$-algebra $R$ is hereditary if its global dimension is at most one, equivalently, $\Ext^2_R(-,-)=0$ for all $R$-modules. The algebra is called tame if, for each $d\geq1$, all but finitely many isomorphism classes of indecomposable $R$-modules of $k$-dimension $d$ occur in one-parameter families. Connected tame hereditary algebras are Morita equivalent to path algebras $kQ$ where $Q$ is an extended Dynkin (Euclidean) quiver: $\widetilde{A}_n$, $\widetilde{D}_n$, $\widetilde{E}_6$, $\widetilde{E}_7$, or $\widetilde{E}_8$ (Hanson et al., 17 Oct 2025). They exhibit infinite representation type with polynomial growth rate $Y_A=1$ (Farnsteiner, 2012).

The Auslander–Reiten quiver of a tame hereditary $R$ splits into three disjoint classes: preprojective modules (ending at projectives under $\tau$), regular modules (arranged in stable tubes), and preinjective modules (ending at injectives under $\tau^{-1}$) (Cadavid et al., 2013).

2. Module Categories and Regular Tubes

For each tame hereditary $R$, the regular part decomposes into a disjoint union of tubes $\mathcal{T}_\lambda$, each of rank $r_\lambda$ (Hügel et al., 2010). Every tube consists of $r_\lambda$ quasi-simple objects on the mouth. Indecomposable regular modules of serial length $n$ are denoted $S[n]$, and infinite-dimensional limits correspond to Prüfer modules $S[\infty]$ and adic modules $S[-\infty]$.

The regular tubes impose constraints on rigid (exceptional) sequences and stratifying systems: in each tube of rank $r$, a partial tilting module or stratifying system may contain at most $r-1$ indecomposable direct summands (Cadavid et al., 2013).

3. Tilting Theory and Universal Localization

A tilting module $T$ over $R$ satisfies $\pdim_R T \leq 1$, $\Ext^1_R(T,T^{(I)})=0$ for all sets $I$, and an exact sequence $0\to R\to T_0\to T_1\to0$ with $T_0,T_1\in \Add T$. Universal localization at a set $\mathcal{U}$ of quasi-simples yields $R_\mathcal{U}$, and forms tilting modules $T_\mathcal{U}=R_\mathcal{U}\oplus(R_\mathcal{U}/R)$. The infinite-dimensional part of any large tilting module is built from such universal localizations and Prüfer modules, with precise classification in terms of the support in regular tubes (Hügel et al., 2010).

Cotilting modules arise as duals of tilting modules, with structure governed by the occurrence of adic and Prüfer summands, and exactly $\mathrm{rank}(\mathcal{T}_\lambda)$ indecomposables in each tube (Hügel et al., 2010).

In the derived category, endomorphism algebras of such tilting modules admit recollements breaking their derived categories into those of $R$ and a product of adèle-type rings $\mathbb{A}_\mathcal{U}$ built from Laurent power series rings $k((x))$, Dedekind domains, and triangular matrix rings. Two distinct finite-length stratifications exist depending on the arrangement of regular cliques: in the Kronecker case, for example, one obtains combinations of derived categories of $k$, $k[x]$, $k((x))$ and Dedekind domains (Chen et al., 2011).

4. Stratifying Systems and Para-Exceptional Sequences

For a family $X=\{X_1,\dots,X_t\}$ of indecomposables over hereditary $A$, $(X_1,\dots,X_t)$ forms a stratifying system (exceptional sequence) if $\Hom_A(X_j,X_i)=0$ for $j>i$ and $\Ext^1_A(X_j,X_i)=0$ for $j\ge i$ (Cadavid et al., 2013). For tame hereditary algebras with $n$ simples, the maximal size of a regular stratifying system is $n-2$; regular-only systems can never be complete.

Recent combinatorics describe para-exceptional sequences: sequences of bricks (endomorphism division algebra) chosen as exceptional objects together with one mouth brick from each non-homogeneous tube. This is essential in realizing the McCammond–Sulway lattice in affine type, reflecting Garside combinatorics not visible in the finite-type noncrossing partition poset. The lattice of para-exceptional subcategories, built by closure-like operations in the module category, is shown to be combinatorial Garside, and lifts the deficiencies of the classical noncrossing partitions in infinite Coxeter types (Hanson et al., 17 Oct 2025).

5. Representation-Theoretic and Geometric Characterizations

Tame hereditary algebras admit equivalent geometric and invariant-theoretic characterizations (Chindris, 2010): for acyclic quivers $Q$, the moduli space $M(Q,\mathbf{d})^{ss}_\theta$ of $\theta$-semi-stable representations of dimension vector $\mathbf{d}$ is a projective space, for all dimension vectors and integral weights. The log-concavity of the sequence $(\dim_k SI(Q,d)_{N\theta})_N$ for all $d$, $\theta$ provides a semi-invariant-theoretic signature of tameness.

Further, the field of rational invariants $k(rep(Q,d))^{GL(\mathbf{d})}$ is either $k$ or $k(t)$ for each Schur root $d$, implying rationality and at most one transcendence in orbit spaces—a property distinguishing tame hereditary and canonical algebras from wild types.

6. Hall Algebras, Rings, and Lie Theoretic Connections

Over finite fields, the Ringel–Hall algebra $\mathscr{H}(A)$ is defined on isoclasses of modules, with multiplication counting extensions. For the duplicated algebra $\bar{A}$, one obtains a triangular decomposition into $$\mathscr{H}(A_0)\,\mathscr{H}(A_{01})\,\mathscr{H}(A_1)$$, mirrored in the composition subalgebra $\mathscr{C}(\bar{A})$. The existence of Hall polynomials $g^M_{X,Y}(x)$ for all indecomposable modules is established in the tame case (Dong et al., 2010).

Upon degeneration ($q=1$), Lie subalgebras $L(\bar{A})$ generated by indecomposables, and $L'(\bar{A})$ generated by the simples, correspond to the positive part of the affine Kac–Moody algebra associated to the extended Dynkin quiver. This exposes deep links between representation theory of tame hereditary algebras and infinite-dimensional Lie theory.

7. Amenability, Polynomial Growth, and Derived Invariants

Amenable representation type, in the sense of Elek, is characterized by hyperfiniteness—every finite-dimensional module decomposes as a large direct sum of small bounded-size summands, with control on the codimension. All tame hereditary algebras over any field are amenable, as are all tame concealed (tilted) algebras (Eckert, 2020, Eckert, 2018). The proof proceeds via induction on tube ranks and defect, using perpendicular categories and reducing to the 2-Kronecker case.

In the homological setting, derived categories of endomorphism algebras from infinite tilting modules over tame hereditary $R$ decompose via recollement into those of $R$ and adèle rings, with distinct finite-length stratifications corresponding to arrangements of simple modules and regular cliques (Chen et al., 2011).

8. Cluster Theory, Maximal Green Sequences, and Mutation

Cluster-tilting theory provides a mutation-based view of the module category of tame hereditary algebras. Maximal green sequences (MGS), chains of green mutations, are finite in number for any tame hereditary algebra, with their lengths forming intervals without gaps—the "No Gap Conjecture" (Hermes et al., 2016). The associated exchange graphs are polygonally connected via finite chains of elementary polygonal deformations, and regular cluster tilting objects mediate the combinatorial geometry of slices and cones in the $c$-vector fan.

For $m$-maximal green sequences in derived categories, each tame hereditary algebra admits only finitely many such mutation sequences for any $m\ge1$, with precise uniform bounds on the number and type of indecomposable summands in intermediate silting objects (Igusa et al., 2017).

9. Extensions, Domesticity, and Group Schemes

Tame hereditary algebras and their radical-square-zero quotients, together with their trivial extensions $A\ltimes D(A)$, precisely parametrize the principal blocks of domestic finite group schemes in odd characteristic. The connection is mediated through split and separable extensions, Morita equivalence, and the geometric structure of binary polyhedral group schemes, leading to explicit presentations for all such blocks (Farnsteiner, 2012).

Summary Table: Key Features of Tame Hereditary Algebras

Feature	Description	Reference
Classification	Path algebras of extended Dynkin quivers	(Hanson et al., 17 Oct 2025)
Module Decomposition	Preprojective, regular (tubes), preinjective	(Cadavid et al., 2013)
Tilting modules	Universal localization, Prüfer modules, Lukas module	(Hügel et al., 2010)
Stratifying systems	Max size $n-2$ for regular summands, never complete	(Cadavid et al., 2013)
Geometric invariants	Projective moduli spaces, log-concave semi-invariants	(Chindris, 2010)
Amenable type	Hyperfinite module decomposition, holds for all tame	(Eckert, 2020)
Hall algebra/affine Lie link	Hall polynomials, PBW bases, Kac–Moody algebra connections	(Dong et al., 2010)
Para-exceptional sequences	Generalization, Garside lattice structure	(Hanson et al., 17 Oct 2025)
Maximal green sequences	Finiteness, no gaps, polygonal connectivity	(Hermes et al., 2016)
Extensions/domestic schemes	Trivial extension structure for group schemes	(Farnsteiner, 2012)

Tame hereditary algebras serve as the paradigm for controlled infinite representation type in finite-dimensional algebra, and as a model of interaction between algebra, combinatorics, geometry, and homological invariants.