Quantum Cluster Algebra with Principal Coefficients

• Quantum cluster algebra with principal coefficients is a noncommutative q-deformation of cluster algebras that integrates frozen variables to yield a universal coefficient structure.
• The algebra employs explicit quantum mutation rules and q-commutation relations that generate quantum Serre-type identities, ensuring structural compatibility and positivity.
• An algebra homomorphism from quantum groups embeds canonical bases and enables categorification via Hall and Hecke algebras, driving applications in Lie theory and integrable systems.

A quantum cluster algebra with principal coefficients is a noncommutative, q-deformation of a cluster algebra in which the initial seed includes, in addition to “cluster variables,” a full set of “frozen” coefficient variables corresponding to principal coefficients. The quantum deformation parameter q is encoded in the commutation relations and exchange relations of the quantum variables. The principal choice of coefficients is structurally fundamental: it yields a universal cluster model, encodes a maximal coefficient structure, and plays a central role in positivity, canonical bases, Hall algebra categorifications, and Lie-theoretic realizations.

1. Definition and Algebraic Framework

Let $\mathcal{A}_q$ denote a quantum cluster algebra with principal coefficients. The construction starts with an initial quantum seed $(\mathbf{X}, B, \Lambda)$, where:

• $$\mathbf{X} = (X_1, \ldots, X_n, X_{n+1}, \ldots, X_{2n})$$ are the quantum cluster and coefficient variables;
• $B \in \operatorname{Mat}_{2n \times n}(\mathbb{Z})$ is the extended exchange matrix, with the lower $n \times n$ block given by an identity matrix (the principal coefficients);
• $$\Lambda \in \operatorname{Mat}_{2n \times 2n}(\mathbb{Z})$$ is a skew-symmetric integer matrix, defining the $q$-commutation relations.

The quantum torus associated to this seed is the $\mathbb{Q}(q^{1/2})$-algebra generated by $\{X_k^{\pm1}\}_{1 \leq k \leq 2n}$, with multiplication

$$X^g X^h = q^{\frac{1}{2} \Lambda(g, h)} X^{g+h}, \quad g,h \in \mathbb{Z}^{2n}.$$

Here, $X^g = \prod_{k=1}^{2n} X_k^{g_k}$. The cluster algebra is generated by iterated quantum mutations, defined by the exchange matrix $B$ and the data of the quantum torus, applying the Berenstein–Zelevinsky quantum mutation rules.

Principal coefficients are those for which the frozen part of the matrix is an identity, so that each exchangeable variable is assigned an auxiliary frozen variable recording the direction of mutation; these frozen variables serve as “universal coefficients.”

2. Quantum Mutation and Commutation Relations

The exchange (mutation) relation for a quantum cluster algebra with principal coefficients is given by

$$X'_k = X^{-e_k} + X^{-e_k + \sum_{i=1}^{2n} [b_{ik}]_+ e_i}$$

where $e_k$ is the $k$-th standard basis vector. The bar-involution (sending $q^{1/2} \mapsto q^{-1/2}$, $X_i \mapsto X_i$) is essential in ensuring that the algebra’s structural constants lie in $\mathbb{Z}[q^{\pm 1/2}]$ and yields a formalism to define bar-invariant and positive bases.

The commutation relations are encoded by $\Lambda$, so that for $1 \leq i, j \leq 2n$,

$X_i X_j = q^{\Lambda_{ij}} X_j X_i.$

Compatible pairs $(B, \Lambda)$ satisfy $B^{\operatorname{T}} \Lambda = \left(\begin{smaLLMatrix} D & 0 \end{smaLLMatrix}\right)$, with $D$ a positive diagonal matrix; this ensures the quantum algebra is well-defined and mutations preserve this compatibility under the standard rules.

3. Fundamental Relations: Quantum Serre-type Identities

A principal structural feature of quantum cluster algebras with principal coefficients is the existence of explicit quantum analogues of the Serre relations among quantum cluster variables generated by mutations from the initial cluster. Let $y_i$ be the quantum cluster variable obtained from mutation in direction $i$ from the initial seed; then for $i \neq j$ and $b_{ij}$ the $ij$-component of $B$:

• If $b_{ij} \leq 0$,

$$\sum_{r=0}^{-b_{ij}+1} (-1)^r q^{d_i \frac{r(r-1)}{2}} \begin{bmatrix} -b_{ij} + 1 \ r \end{bmatrix}_{q^{d_i}} y_i^{-b_{ij}+1-r} y_j y_i^r = 0,$$

• If $b_{ij} > 0$,

$$\sum_{r=0}^{b_{ij}+1} (-1)^r q^{d_i \frac{r(r-1)}{2} - r b_{ij}} \begin{bmatrix} b_{ij} + 1 \ r \end{bmatrix}_{q^{d_i}} y_i^{b_{ij}+1-r} y_j y_i^r = 0,$$

where $d_i$ is the skew-symmetrizer and $$\begin{bmatrix} n \ r \end{bmatrix}_{q^d}$$ denotes the $q$-binomial coefficient. These identities generalize the quantum Serre relations of $U_q(\mathfrak{n})$ and reflect a deep structural compatibility between the quantum cluster algebra and the corresponding quantum group (Huang et al., 15 Sep 2025).

Furthermore, higher-order versions involving positive powers of variables and extra multipliers can be derived by similar recursive and combinatorial arguments.

4. Homomorphism from the Quantum Group

A major consequence of the fundamental relations is the immediate construction of an algebra homomorphism from the (untwisted) quantum group $U_q(\mathfrak{n})$ associated to the Cartan matrix $C$ (with $c_{ii} = 2$, $c_{ij} = -|b_{ij}|$ for $i \neq j$) into the quantum cluster algebra $\mathcal{A}_q$. Specifically, mapping the Chevalley generators $\theta_i$ to the quantum cluster variables $y_i$ for all $1 \leq i \leq n$,

$\theta_i \mapsto y_i,$

extends to an algebra homomorphism

$\widehat{U}^+(C, q) \to \mathcal{A}_q,$

since the $y_i$ satisfy the same quantum Serre relations as the $\theta_i$. This establishes a structural embedding of the quantum group into $\mathcal{A}_q$, positioning the cluster algebra as a natural home for phenomena from quantum groups, canonical bases, and categorification theory (Huang et al., 15 Sep 2025).

5. Canonical Bases, Duality, and Categorification

Quantum cluster algebras with principal coefficients admit (and in many cases, canonically determine) families of bases with positivity and bar-invariance properties. Notably:

• Atomic (i.e., canonically positive) bases based on the dual canonical and PBW bases of the corresponding positive parts of quantum groups;
• Distinguished bar-invariant bases parameterized by Chebyshev polynomials or quantum Caldero–Chapoton characters, specialized to the Kronecker and affine types (Ding et al., 2021, Ding et al., 2020).

These bases are constructed using recursive formulas, often quantizing known relations in the commutative setting, and their structure constants are confirmed to be positive Laurent polynomials over $\mathbb{Z}[q^{\pm1/2}]$ when expanded in any cluster. The cluster monomials and, in affine types, suitable combinations with quantum Chebyshev polynomials, generate these bases concretely.

Categorification approaches—via Hall algebras of quiver representations, their morphism categories, and the associated bialgebra structures—realize these bases as images of (quantum) cluster characters, and identify the multiplication structure of the cluster algebra with extension data in the category (Ding et al., 2019, Fu et al., 2020). The associated positivity, and identification of canonical bases with dual semicanonical or Kazhdan–Lusztig bases, is both structurally and representation-theoretically significant.

6. Applications and Structural Implications

• The fundamental Serre-type relations control the structure, mutations, and Laurent expansions of quantum cluster variables in the algebra, providing direct calculational tools.
• The surjective quantum group homomorphism makes quantum cluster algebras natural recipients for canonical bases, total positivity, and related algebraic structures.
• In Lie theory, quantum cluster algebra structures have been established on quantized coordinate rings—quantum double Bruhat cells and quantum unipotent rings—proving the Berenstein–Zelevinsky conjecture and unifying the algebraic, combinatorial, and categorification perspectives (Goodearl et al., 2016, Goodearl et al., 2015).
• For acyclic seeds, the new lower bound quantum cluster algebra generated by initial and quantum projective cluster variables coincides with the full quantum cluster algebra and admits a dual PBW basis, paralleling the classic situation (Huang et al., 25 Jul 2024).
• In generalized and non–simply-laced types (skew-symmetrizable), the quantum virtual Grothendieck ring and associated quantum cluster algebra structures, together with braid-move isomorphisms and categorification by quiver Hecke algebras, demonstrate quantum Laurent positivity and deep dualities (Jang et al., 2023, Lee et al., 13 Feb 2024).
• The combinatorial separation of variables and explicit cluster multiplication formulas, such as in affine type $A_2^{(1)}$ (Yang et al., 14 Apr 2025), and for T-systems with principal coefficients (Vichitkunakorn, 2015), illustrate the richness of the principal coefficient framework.

7. Summary Table: Fundamental Relations, Homomorphisms, and Basis Structure

Structural Feature	Formula / Principle	Reference
Quantum Serre relation (b_{ij}≤0)	$$\sum_{r=0}^{-b_{ij}+1} (-1)^r q^{d_i\frac{r(r-1)}{2}} \binom{-b_{ij}+1}{r}_{q^{d_i}} y_i^{-b_{ij}+1-r} y_j y_i^r = 0$$	(Huang et al., 15 Sep 2025)
Algebra homomorphism $U_q^+ \to \mathcal{A}_q$	$\theta_i \mapsto y_i$; $\mathcal{A}_q$ satisfies quantum Serre relations	(Huang et al., 15 Sep 2025)
Canonical / atomic basis	Bar-invariant, positive, and contains all monomials in clusters and dual Chebyshev polynomials	(Ding et al., 2021, Ding et al., 2020)
Hall algebra categorification	Quantum cluster algebra is a subquotient of localized Hall algebra	(Ding et al., 2019, Fu et al., 2020)
Lower bound / projective generation	Quantum projectives + initial cluster variables generate acyclic quantum cluster algebra	(Huang et al., 25 Jul 2024)

8. Concluding Perspective

Quantum cluster algebras with principal coefficients are characterized by explicit quantum Serre-type relations among variables arising from mutations, admitting canonical bar-invariant bases and positive Laurent phenomenon. The framework realizes embeddings of quantum groups, supports categorification by Hall and Hecke algebras, and structures a broad array of applications in representation theory, Lie theory, and integrable systems. The explicit and universal nature of principal coefficients makes these algebras foundational in the quantum cluster theory landscape.