Silted Algebras: Classification & Structure

• Silted algebras are defined as the endomorphism algebras of 2-term silting complexes over hereditary algebras, bridging classical tilting and strictly shod algebras.
• They exhibit a distinctive homological property where every indecomposable module has either projective or injective dimension at most one, with a global dimension of at most three.
• Combinatorial and geometric techniques classify silted algebras for Dynkin types A and D, providing explicit enumeration and structural insights.

A silted algebra is defined as the endomorphism algebra of a 2-term silting complex in the bounded homotopy category of projectives over a hereditary algebra, or more generally as an algebra derived equivalent to such an endomorphism algebra. Silted algebras serve as a cornerstone in the modern generalization of classical tilting theory, encapsulating both tilted and strictly shod algebras—those whose indecomposable modules all have either projective or injective dimension at most one, and where global dimension is at most 3. Recent developments have produced precise classification results and combinatorial formulae for silted algebras associated with Dynkin quivers of types A and D, including explicit enumeration and geometric/topological characterizations. This entry provides a comprehensive survey of the foundational concepts, structural results, and classification techniques underpinning silted algebras, with an emphasis on the distinctive homological and combinatorial features as established in the recent literature.

1. Definition, Characterizations, and General Homological Properties

A finite-dimensional algebra $B$ is called silted if there exists a hereditary finite-dimensional algebra $A$ and a 2-term silting complex $P$ in $K^b(\mathrm{proj}\,A)$ such that $B \cong \operatorname{End}_{D^b(A)}(P)$. A complex $P$ is 2-term silting if it is concentrated in degrees $0$ and $-1$, satisfies $\mathrm{Hom}(P, P[1])=0$, and generates the triangulated category. The concept generalizes that of tilted algebras (endomorphism algebras of tilting modules over hereditary algebras) and strictly shod algebras (which are not tilted and have global dimension 3 but every indecomposable module has projective or injective dimension at most one) (Buan et al., 2015).

The notion of 'quasi-silted' extends this definition to Ext-finite hereditary abelian categories $\mathcal{H}$ (not necessarily module categories), where $B = \operatorname{End}_{D^b(\mathcal{H})}(P)$ for a 2-term silting complex $P$ is called quasi-silted; in this case, all strictly shod algebras are quasi-silted and vice versa (Buan et al., 2015).

The defining homological property of a silted algebra $B$ is that for every indecomposable $B$-module $X$, either $\mathrm{pd}_B X \leq 1$ or $\mathrm{id}_B X \leq 1$. The global dimension satisfies $\mathrm{gldim}(B) \leq 3$. The following table summarizes the defining homological distinctions:

Algebra type	Homological condition	Global dim.
Tilted	$\mathrm{pd}_B X \leq 1$ $\forall X$ or $\mathrm{id}_B X \leq 1$ $\forall X$	$\leq 2$
Strictly shod	For each $X$, $\mathrm{pd}_B X \leq 1$ or $\mathrm{id}_B X \leq 1$	$=3$
Silted	As above (tilted $\cup$ strictly shod)	$\leq 3$

Silted algebras also arise as the hearts of certain $t$-structures in $D^b(A)$ and are closely tied to torsion pairs induced by the silting complex (Buan et al., 2015).

2. Structure via Silting Complexes and Torsion Pairs

Given a 2-term silting complex $P \in K^b(\mathrm{proj}\,A)$, there is an associated triangle: $H^{-1}(P)[1] \to P \to H^0(P) \to H^{-1}(P)[2]$ Here, $H^0(P)$ and $H^{-1}(P)[1]$ determine complementary 'torsion' and 'torsion-free' classes in $\mathrm{mod}\,B$ via functors $X(P) = \operatorname{Hom}_{D^b(A)}(P, F(P)[1])$ and $Y(P) = \operatorname{Hom}_{D^b(A)}(P, T(P))$, with $$T(P) = \{ M \mid \operatorname{Hom}_{D^b(A)}(P,M[1])=0 \}$$ and $$F(P) = \{ M \mid \operatorname{Hom}_{D^b(A)}(P, M)=0 \}$$. These define a split torsion pair in $\mathrm{mod}\,B$ and induce a bounded $t$-structure whose heart is $\mathrm{mod}\,B$ (Buan et al., 2015). As a result, the representation theory of a silted algebra is encoded in the combinatorics of its silting complexes and the associated $t$-structures.

3. Classifications for Quivers of Dynkin Type A and D

Type $\mathbb{A}_n$

For linearly oriented type $\mathbb{A}_n$ quivers, the classification is complete: every silted algebra is either a tilted algebra of $\mathbb{A}_n$ or a direct product of two smaller tilted algebras $A_m \times A_{n-m}$, $1 \leq m \leq n-1$. No strictly shod silted algebras arise in this setup (Liu et al., 2022, Zhang et al., 2022, Xie et al., 21 Apr 2025).

Algorithmically, all 2-term silting complexes are described by combinatorial operations involving partitions of the quiver and subsequent gluings of rooted subquivers representing tilting modules over smaller path algebras (Xie et al., 21 Apr 2025, Xing, 2021). The number of silted algebras of type $\mathbb{A}_n$ is hence expressible via Catalan numbers and associated convolutions: $$\begin{aligned} \operatorname{as}(A_n) &= \operatorname{at}(A_n) + \sum_{m=1}^{n-1} \operatorname{at}(A_m) \cdot \operatorname{at}(A_{n-m}) \ \operatorname{at}(A_n) &= \text{number of tilted algebras of type } A_n \end{aligned}$$ with $\operatorname{at}(A_n)$ given by the Catalan sequence (Liu et al., 2022, Xie et al., 21 Apr 2025).

Type $\mathbb{D}_n$

For Dynkin $\mathbb{D}_n$ quivers (for $n \geq 5$ with both linear and mutated orientations), silted algebras decompose into several explicit families as unions of products of tilted algebras corresponding to subquivers—that is, combinations such as $\mathfrak{t}(\Lambda_n)$ (tilted algebras of certain types), direct products like $$\mathfrak{t}(\Lambda_m) \times \mathfrak{t}(A_{n-m})$$, and strictly shod algebras distinguished by effective intersection criteria in the structure of the tilting module decomposition (Zhang, 7 Sep 2025).

The strictly shod silted algebras in this setup are precisely characterized and enumerated: $$a_{ss}(\Lambda_n) = a_{nht}(A_{n-1}) - 2a_{nht}(A_{n-2}) = \frac{1}{n} \binom{2n-2}{n-1} - \frac{2}{n-1} \binom{2n-4}{n-2}$$ where $a_{nht}(A_m)$ is the number of non-hereditary tilted algebras of $A_m$ (Zhang, 7 Sep 2025).

4. Strictly Shod Silted Algebras and String Algebra Structure

A central finding for Dynkin type $\mathbb{D}_n$ is that all strictly shod silted algebras are string algebras. String algebras are defined by presentation conditions: at most two arrows at each vertex, and relations such that nonzero paths of length two are highly constrained (at most one nontrivial composition at each relevant vertex, and no overlapping). This makes their homological properties and indecomposable module categories explicitly computable and accessible to combinatorial methods (Zhang, 7 Sep 2025).

Both the quiver orientation and the combinatorial structure of the tilting module decomposition fully determine the string algebra presentation for strictly shod cases. This observation provides a concrete combinatorial handle for classifying the representation theory of all strictly shod silted algebras arising in this context.

5. Counting Isomorphism Classes: Explicit Formulas

The combinatorial methods developed in the aforementioned classifications lead to closed formulae for the number of silted and strictly shod algebras of Dynkin type: $$a_{ss}(\Lambda_n) = \frac{1}{n} \binom{2n-2}{n-1} - \frac{2}{n-1} \binom{2n-4}{n-2}$$ for strictly shod algebras of type $\Lambda_n$ (one orientation of $\mathbb{D}_n$) and analogously for the total number of silted algebras via sums and products over smaller types. These formulas are derived from the combinatorics of tilting submodules, rooted quivers, and their gluings, with the Catalan numbers playing a central role (Xie et al., 21 Apr 2025, Zhang, 7 Sep 2025).

6. Geometric Realizations

For algebras of type $\mathbb{A}_n$ and certain gentle/quasi-gentle hereditary types, geometric models play a critical role in classification and understanding. In these models, triangulations of marked ribbon surfaces correspond to collections of silting objects, with combinatorial moves such as flips and cuts realizing silting mutations and reductions. The absence of strictly shod silted algebras in hereditary gentle types is a topological reflection of the polygonal decomposition and the constraint that every tile in the associated surface is a triangle (no polygons with more than three edges) (Liu et al., 2022, Zhang et al., 2022, Chang et al., 2020).

7. Realization Functor Obstructions and Derived Equivalence

Even when a silted algebra $B$ is derived equivalent to its hereditary progenitor $A$, the realization functor from the heart of the $t$-structure induced by a silting complex to the derived category $D^b(\mathrm{mod}\,A)$ does not always extend to a derived equivalence. There exist explicit families—constructed via particular shifts and summands in the silting complex—where the embedding of the heart is not extensible, obstructed by the vanishing (or not) of certain higher Yoneda Ext groups. These serve as counterexamples to naive expectations of equivalence extension and sharply delineate the necessary conditions (Zhang, 7 Sep 2025).

8. Open Directions and Further Implications

The classification methodology—comprising the combinatorics of tilting decompositions, string algebra analysis, and geometric interpretation—opens the possibility of extending these results to arbitrary orientations, higher types, or settings incorporating more general silting and support $\tau$-tilting theory. A principal open question addressed is whether every strictly shod algebra of Dynkin type $\mathbb{D}_n$ (for all orientations) is necessarily a string algebra, a conjecture supported by all evidence in the classified cases (Zhang, 7 Sep 2025). Further applications to non-compact tilting theory, triangulated category reductions, and the full invariance theory of silting-discrete and derived-discrete classes are suggested by connections to recent studies (Aihara et al., 2023).

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Silted algebras thus form an essential bridge between homological algebra and the combinatorial structure of derived and module categories. The intricate connections between their mutation classes, geometrization, and derived invariants continue to drive advances in modern representation theory.