Rooted Cluster Morphisms

• Rooted cluster morphisms are structure-preserving maps between rooted cluster algebras that maintain the initial seed and ensure compatibility with all admissible mutation sequences.
• They integrate combinatorial, algebraic, and categorical frameworks, facilitating applications to surface models, representation theory, and symmetry analysis in cluster theory.
• Their classification into ideal, inducible, and partial variants underpins detailed studies of subalgebras, tensor decompositions, and topological modifications in cluster algebra research.

A rooted cluster morphism is a structure-preserving homomorphism between rooted cluster algebras and serves as a crucial concept integrating the combinatorial, algebraic, and categorical frameworks underlying cluster theory. Rooted cluster morphisms require not only preservation of algebraic structure but full compatibility with the rooted seed and all admissible mutation sequences. The maturation of this notion has enabled sophisticated generalizations—connecting cluster algebras to surface models, representation theory, and the broader landscape of cluster combinatorics—while providing a precise categorical language for morphisms beyond invertible symmetries.

1. Definition and Core Properties

A rooted cluster morphism is defined for rooted cluster algebras, i.e., cluster algebras endowed with an explicit initial ("rooted") seed $\Sigma = (X, ex, B)$, where $X$ is the cluster, $ex$ is the set of exchangeable variables, and $B$ is the exchange matrix.

A ring homomorphism $f: \mathcal{A}(\Sigma) \to \mathcal{A}(\Sigma')$ is a rooted cluster morphism if it satisfies three structural axioms (see (Gratz, 2014, Chang et al., 2014)):

• (CM1): $f(X) \subseteq X' \cup \mathbb{Z}$, i.e., each cluster variable from $X$ is sent to a cluster variable of the target, or to an integer (specialization).
• (CM2): $f(ex) \subseteq ex'$, i.e., exchangeable (mutable) variables are mapped to exchangeable variables.
• (CM3): $f$ commutes with mutation along any biadmissible sequence: for any $E$-admissible mutation sequence $(x_1, ..., x_n)$ in $\Sigma$, if $(f(x_1), ..., f(x_n))$ is $E'$-admissible in $\Sigma'$, then

$$f(\mu_{x_n} \circ ... \circ \mu_{x_1}(y)) = \mu_{f(x_n)} \circ ... \circ \mu_{f(x_1)}(f(y)), \quad \forall y \in X.$$

This "commutation with mutation" is the central structural constraint. The seed data—especially the rooted seed—remains privileged throughout, rendering these morphisms "rooted" in both a strict and categorical sense.

Rooted cluster morphisms generalize cluster automorphisms (Assem et al., 2010), which are invertible morphisms sending clusters to clusters and commuting with all mutations—thus, automorphisms are invertible rooted cluster morphisms, i.e., isomorphisms in the category of rooted cluster algebras.

2. Classification, Ideal Morphisms, and Partial Variants

Rooted cluster morphisms formalize the notion of morphisms in the "category Clus" of rooted cluster algebras (Gratz, 2014, Chang et al., 2014). A rooted cluster morphism $f$ is ideal if its image is the cluster algebra generated by the image of the initial seed: $f(\mathcal{A}(\Sigma)) = \mathcal{A}(f(\Sigma)).$ A key result is that any rooted cluster morphism without specializations (i.e., all cluster variables map to cluster variables, not integers) is ideal, but rooted cluster morphisms with specializations may not be ((Gratz, 2014), Proposition 3.27; (Chang et al., 2014), Examples 3–4).

Further, inducible morphisms—those lifting to ambient (localized polynomial) ring homomorphisms—are ideal if and only if they can be factored as a surjective composed with an injective rooted cluster morphism. Injective morphisms are characterized as inclusions arising from freezing subsets of exchangeable variables, and are sections if and only if no extra freezing occurs ((Chang et al., 2014), Theorem 2.23).

Partial seed homomorphisms and partial ideal rooted cluster morphisms are defined on subalgebras associated to mixing-type sub-seeds (Huang et al., 2016). These constructions allow a fine-grained correspondence between sub-rooted cluster algebras and regular $\mathcal{D}$-classes in the semigroup of partial seed endomorphisms, with applications to internal classification and geometry of subalgebras.

3. Combinatorial and Categorical Structures

The combinatorics of rooted cluster morphisms interlace with quiver mutations, mixing-type sub-seeds, and gluing/specialization operations (Huang et al., 2015, Huang et al., 2016). A rooted cluster subalgebra corresponds to a mixing-type sub-seed $\Sigma_{I_0, I_1}$—obtained by freezing variables in $I_0$ and deleting those in $I_1$—if and only if the induced exchange matrix allows for compatible mutation patterns after the modification ((Huang et al., 2015), Theorem 4.14). Surjective rooted cluster morphisms decompose via canonical gluing of frozen variables and subsequent isomorphisms.

These operations possess explicit categorical and combinatorial characterizations:

• The tensor decomposition of a rooted cluster algebra as a quotient of tensor products of algebras associated to indecomposable components ((Chang et al., 2014), Theorem 2.20).
• Monoidal categorification of rooted subalgebras: a rooted cluster subalgebra inherits a monoidal categorification from its ambient algebra ((Huang et al., 2015), Theorem 5.6).

Geometric interpretations further relate sub-rooted cluster algebras to "paunched surfaces" in the context of cluster algebras from surfaces—cutting arcs (freezing or deleting cluster variables) corresponds to explicit modifications of the underlying surface, yielding a bijection between subalgebras and homeomorphism classes of paunched surfaces ((Huang et al., 2016), Theorem 6.9).

4. Representation-Theoretic and Homological Connections

Rooted cluster morphisms are connected to representation theory via categorifications associated with $2$-Calabi–Yau triangulated categories and cluster categories (Chang et al., 2014). In this setting, cotorsion pairs in such a category correspond bijectively to "complete pairs" of rooted cluster subalgebras—subalgebra pairs characterized by mutual orthogonality conditions and rigid subcategories ((Chang et al., 2014), Theorem 3.13). This correspondence refines the link between algebraic and categorical structures, with t-structures mapping to complete pairs in the coefficient-free case.

Tensor decompositions of rooted cluster algebras also reflect homological decompositions in the underlying categorifications, elucidating how algebraic and categorical data are constructed from indecomposable objects and their endomorphism structure.

5. Group-Theoretic and Symmetry Aspects

Rooted cluster morphisms are intricately related to the automorphism and symmetry groups of cluster algebras. Cluster automorphisms (Assem et al., 2010) are invertible rooted cluster morphisms whose group is often computable via:

• The automorphism group of the transjective component in the cluster category for acyclic algebras (through a surjective group homomorphism, Theorem 3.8).
• Mapping class groups for cluster algebras from surfaces, with injective homomorphisms into the cluster automorphism group ((Assem et al., 2010), Theorem 4.11).

Explicit computations in Dynkin and Euclidean types provide group-theoretic invariants (e.g., finite cyclic, or more complex semidirect product presentations) that classify cluster automorphism types and potentially restrict the range of rooted cluster morphisms in automorphism-finite contexts ((Assem et al., 2010), Theorem 5.1).

6. Structural, Topological, and Geometric Consequences

The category Clus of rooted cluster algebras and rooted cluster morphisms admits rich structure:

• Rooted cluster algebras of infinite rank are colimits (linear unions) of finite rank ones, facilitating the transfer of properties such as positivity ((Gratz, 2014), Theorem 4.6 and 4.9).
• Partial seed homomorphisms and regular $\mathcal{D}$-classes provide discrete invariants for classifying sub-structures (Huang et al., 2016).
• Frozenization, gluing, and specialization yield explicit classification and enumeration results for rooted cluster subalgebras and quotient algebras (Huang et al., 2015).
• In surface models, modifications correspond to clear topological operations on Riemannian surfaces, making the theory accessible to geometric and topological methods.
• Representation-theoretic perspectives connect morphisms to t-structures, cotorsion pairs, and categorified invariants within 2-Calabi–Yau settings.

7. Open Problems and Future Directions

Important open problems include the full classification of ideal rooted cluster morphisms (with and without specializations), the extension of colimit constructions to the uncountable setting, further elucidation of automorphism finiteness and its relationship to rooted morphism groups, and the explicit realization of combinatorial and geometric models (e.g., paunched surfaces) in higher or non-acyclic settings (Gratz, 2014, Huang et al., 2015, Huang et al., 2016). The ongoing unification of categorical, combinatorial, and geometric approaches continues to inform the understanding and application of rooted cluster morphisms in related domains.

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In sum, rooted cluster morphisms form the categorical scaffold connecting the combinatorial, geometric, and representation-theoretic aspects of cluster algebras, generalizing invertible symmetries, refining structural decompositions, and categorifying the mutation dynamics central to cluster theory (Assem et al., 2010, Gratz, 2014, Chang et al., 2014, Huang et al., 2015, Huang et al., 2016).