Abstract Quantum Cluster Structures

• Abstract quantum cluster structures are a categorical formalism that encapsulates mutation and quasi-commutation combinatorics in both classical and quantum settings.
• They enable a precise passage between tropical and quantum frameworks by using functorial mappings between exchange graphs, lattices, and quantum tori.
• The theory unifies diverse realizations—from cluster algebras and varieties to triangulation models—while offering new insights into representation theory and categorification.

An abstract quantum cluster structure is a categorical and combinatorial formalism designed to encode the mutation and quasi-commutation combinatorics intrinsic to quantum cluster algebras and their diverse geometric and representation-theoretic avatars. This concept organizes the underlying mutation data, independent of any specific coordinate realization, in a manner that allows a precise passage between classical (tropical) and quantum frameworks and enables functorial relationships between different types of cluster-theoretic objects. The theory facilitates comparison and interrelation among quantum cluster algebras, cluster varieties, cluster categories, and triangulation models at a universal combinatorial level (Grabowski et al., 3 Oct 2025).

1. Foundational Data and Tropical Level Abstraction

An abstract cluster structure in the classical (unquantized) setting is defined by the following categorical package:

• A directed graph $E$ (often interpreted as a signed path category) whose vertices $C$ encode clusters or seeds.
• Two contravariant functors (sheaves of Abelian groups) $X: E^\mathrm{op} \to \mathrm{Ab}$ and $A: E \to \mathrm{Ab}$, where:
  • $X(c)$ plays the role of the “$X$-lattice” (e.g., $\mathbb{Z}^n$ for a cluster with $n$ mutable variables),
  • $A(c)$ is the free Abelian group with basis the cluster variables at $c$.
• A factorization (boundary map) ${}_c: X(c) \to A(c)$ for each vertex $c$, subject to a naturality condition: for any mutation/edge $f: c\to d$,

$A(f) \circ {}_c = {}_d \circ X(f).$

• A canonical evaluation pairing

$$\langle a, x \rangle = x(a), \quad \text{for } a \in A(c), \; x \in X(c),$$

which recovers, in concrete models, the exchange matrix or its tropical version.

The entire mutation structure is encoded at the level of these functors and their transforms, abstracting away from explicit coordinate formulas. This approach underlies the ability to represent cluster algebras, cluster varieties, and more general combinatorial models such as surface triangulations purely at the tropical/combinatorial level.

2. Quantum Structures and Additional Data

To define an abstract quantum cluster structure, the framework is upgraded by enriching the Abelian group data with quantum information:

• For each $c \in E$, the free Abelian group $A(c)$ is “exponentiated” to produce a quantum torus

$T_q(B) = (K[B])^{\Omega_q},$

where the twisted multiplication is determined by a bicharacter

$$x^v x^w = q^{\frac{1}{2}\langle v, w \rangle} x^{v+w}$$

with the pairing $\langle -, - \rangle$ induced by the underlying skew-symmetric data (typically arising from a compatible pair $(B,L)$ in the language of quantum cluster algebras).

• Classical/quantum compatibility is implemented via the choice of a “retraction” (splitting) $\rho: A \to X$, satisfying $\rho \circ {}_c = \mathrm{id}_{X(c)}$ for each $c$.
• The quantum datum is then the composition

$\epsilon = \delta_X \circ \rho,$

with $\delta_X$ the canonical isomorphism associated to the skew-symmetric form.

• The critical compatibility condition is that the adjoint of the factorization, taken with respect to the evaluation pairing, satisfies $({}_c)^* = -{}_c$.

This quantum data captures, at the categorical level, the information governing quasi-commutation relations and quantum mutation, abstracting the Berenstein–Zelevinsky compatibility conditions from explicit matrices to functorial and natural transformation language.

3. Morphisms and the Category of Abstract Quantum Cluster Structures

A core innovation is the introduction of a well-behaved category $\mathcal{QC}$ whose objects are abstract quantum cluster structures and whose morphisms are triples $(F,\chi,\alpha)$, where:

• $F: E_1 \to E_2$ is a functor between the underlying exchange graphs/path categories,
• $\chi: X_1 \to X_2 \circ F^{\mathrm{op}}$ and $\alpha: A_1 \to A_2 \circ F$ are natural transformations between the $X$ and $A$ sheaves, respectively.

Compatibility requires commutativity of

$\begin{tikzcd} X_1(c) \arrow[r, "{\,_{1,c}}"] \arrow[d, "\chi_c"'] & A_1(c) \arrow[d, "\alpha_c"] \ X_2(F(c)) \arrow[r, "{\,_{2,F(c)}}"'] & A_2(F(c)) \end{tikzcd}$

for all $c$. This structure admits constructions of initial and terminal objects, products, and coproducts, and encompasses rooted cluster morphisms as special cases. Notably, morphisms in this category formalize how different realizations (algebraic, geometric, and categorical) with the same underlying abstract combinatorics are related.

4. Relationship with Quantum Cluster Algebras, Varieties, and Categories

Abstract quantum cluster structures are related to concrete quantum cluster objects via “exponentiation–tropicalization” duality:

• Any quantum cluster algebra (as per Berenstein–Zelevinsky) tropicalizes to an abstract quantum cluster structure by logarithmic passage from the quantum torus coordinates to the group data $(X,A)$ and their pairings.
• Conversely, exponentiating the underlying group data of an abstract quantum cluster structure recovers the associated quantum torus, and, with mutation data, the corresponding quantum cluster algebra (Grabowski et al., 3 Oct 2025).

This categorical abstraction allows comparison of, for example, the cluster algebra structure on the coordinate ring $\mathcal{O}(\mathrm{Gr}(2,6))$ and the geometric model from triangulations of the hexagon: both admit isomorphic abstract quantum cluster structures, even when direct algebraic realization maps are unavailable.

The formalism is also applicable to:

• Quantum cluster varieties: gluing split quantum tori via functors such as $\operatorname{Hom}(-, \mathbb{G}_m)$.
• Cluster categories: where $A(T) = K_0(T)$ and $X(T) = K_0(\mathrm{fd}\,T)$ for cluster tilting subcategories $T$, with the factorization given by the “index” or “cluster character.”

5. Canonical Examples and Illustrative Case Studies

The theory encompasses and systematizes a broad diversity of quantum cluster phenomena:

Realization Type	$A(c)$ (Label Group)	$X(c)$ (Tropical Lattice)
Quantum cluster algebra	$\mathbb{Z}^m$	$\mathbb{Z}^n$
Triangulated surface model	Free on arcs	Free on quadrilaterals
Cluster category (2-CY)	$K_0(T)$	$K_0(\mathrm{fd}\,T)$
Quantum Grassmannian ring	Plücker variables	Relations from flips/quads

For each, the abstract quantum cluster structure records the same mutation combinatorics (at the quantum level) and underpins equivalences and embeddings between these different models at the level of cluster combinatorics.

A concrete textbook example is the Grassmannian $\mathrm{Gr}(2,6)$. The cluster algebra on its homogeneous coordinate ring, as well as the triangulation model from the hexagon, give rise to isomorphic abstract quantum cluster structures (with explicit morphisms constructed via matching labelling data and natural transformations).

6. Structural and Categorical Implications

The abstract quantum cluster structure encapsulates mutation and quantum deformation as categorical data, offering several advantages:

• It isolates the fundamental mutation combinatorics and quasi-commutation relations from concrete presentations.
• The existence of a universal category of such structures enables unification of different cluster-theoretic frameworks.
• Rooted cluster morphisms and other inter-model relationships lift naturally to morphisms in this setting.
• It clarifies which features are genuinely invariant (for example, combinatorics of the exchange graph) versus those that are model-dependent (coordinate/algebraic realization, frozen variables, etc.).
• In the quantum context, the formalism clarifies how extra structure (for example, retraction and symmetry of the quantum datum) enters categorically, and how the quantum and classical categories differ.

7. Connection to Representation Theory and Future Directions

Categorical abstractions of quantum cluster structure align naturally with recent advances in categorification and representation theory. The existence of canonical quantum data, indexed by Grothendieck groups and related to exchange matrices, surfaces directly in representation-theoretic categorification of cluster algebras via categorified cluster characters, 2-Calabi–Yau categories, and monoidal categorifications (Grabowski et al., 18 Nov 2024).

Potential directions include:

• Systematic classification of abstract quantum cluster structures of small (or infinite) rank.
• Formal paper of how quantum deformation parameters affect functoriality at the abstract level.
• Integration with the combinatorics of categorified canonical and cluster bases, especially in the context of quantum unipotent groups and quantum affine algebra representations.

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This approach thus packages the essential mutation and quantum deformation data in a universal, categorical, and functorial manner, providing a foundational language in which all quantum cluster–theoretic phenomena can be compared and transferred across algebraic, geometric, and categorified settings.