2-Parameter Deformed Cartan Matrices

• 2-Parameter Deformed Cartan Matrices is a framework that embeds indeterminates q and t into symmetrizable Cartan matrices, integrating classical root-theoretic and quantum algebraic data.
• The deformation employs combinatorial braid-group operators and graded Euler–Poincaré theory to yield categorical interpretations in module categories and quiver varieties.
• Applications include monoidal categorifications, Langlands duality correspondences, and insights into the representation theory of quantum affine algebras and generalized preprojective algebras.

A two-parameter deformation of Cartan matrices introduces indeterminates $(q,t)$ into the structure of symmetrizable generalized Cartan matrices, yielding families of matrices $C(q,t)$ whose entries encode both classical root-theoretic and quantum algebraic data. This deformation emerges naturally in the study of quantum affinizations, monoidal categorifications, and the representation theory of generalized preprojective algebras, particularly in the context of Langlands duality and the theory of quiver $\mathcal{W}$-algebras. The deformation admits detailed combinatorial and categorical interpretations, notably through the action of braid-group operators and via the graded Euler–Poincaré theory on module categories attached to quivers. These insights, developed by Fujita and Murakami across finite and general type cases, establish deep connections to quantum affine algebras, the geometry of quiver varieties, and conjectural correspondences in tensor product categorifications (Fujita et al., 2021, Fujita et al., 2023).

1. Construction of the $(q,t)$-Deformed Cartan Matrix

Given a symmetrizable generalized Cartan matrix $C=(c_{ij})_{i,j\in I}$ with minimal left symmetrizer $D = \mathrm{diag}(d_i)_{i \in I}$, and indeterminates $q,t$, the $(q,t)$-deformed Cartan matrix $C(q,t)$ is defined by

$$C_{ij}(q,t) = \begin{cases} q^{d_i}t^{-1} + q^{-d_i}t, & i = j, \ [c_{ij}]_q, & i \ne j, \end{cases}$$

where $$[k]_q = (q^k - q^{-k})/(q - q^{-1}) \in \mathbb Z[q^{\pm 1}]$$ (Fujita et al., 2021).

In the general type setting, for $i\sim j$ adjacent in the Dynkin diagram, introduce $g_{ij} = \gcd(|c_{ij}|, |c_{ji}|)$ and $f_{ij} = |c_{ij}|/g_{ij}$, then set

$$C_{ij}(q,t) = \begin{cases} q^{d_i}t^{-1} + q^{-d_i}t, & i = j, \ -\delta_{i\sim j} g_{ij} [f_{ij}]_{q^{d_i}}, & i \ne j, \end{cases}$$

with $[m]_x = (x^m - x^{-m})/(x - x^{-1})$ (Fujita et al., 2023).

The deformed Cartan matrix specializes to $C$ for $(q,t) = (1,1)$ and, for all types, yields an invertible matrix over $\mathbb Z[q^{\pm 1}, t^{\pm 1}]$.

2. Combinatorial and Braid-Group Formulation

No canonical closed-form for $C(q,t)$ exists across all types; instead, the entries can be expressed combinatorially via braid-group operators:

• Fix a (possibly infinite) reduced sequence $(i_1, i_2, \dots)$ adapted to the Weyl group.
• Associate to each simple root $i$ a $Q$-linear deformed reflection

$T_i(x) = x - q^{-d_i} t C_{ij}(q,t) \alpha_i$

(for $x$ in the root space), which satisfy the braid relations (Fujita et al., 2021, Fujita et al., 2023).

• For dual basis elements $\{\varpi_i^\vee\}$, the $(i,j)$ entry is

$$C_{ij}(q,t) = q^{d_j} t^{-1} \sum_{k>0,\, i_k = j} (\varpi_i^\vee, T_{i_1}^{-1} \cdots T_{i_{k-1}}^{-1}(\alpha_j))_{q,t}$$

in the finite case, or its generalization for arbitrary type.

• The inverse matrix admits the expansion

$$C(q,t)^{-1} = q^D t^{-1} (\mathrm{id} + A(q,t) + A(q,t)^2 + \dots)$$

with $A(q,t) = \mathrm{id} - q^D t^{-1} C(q,t)$ (Fujita et al., 2021).

3. Interrelation with Generalized Preprojective Algebras

The categorical interpretation relies on the generalized preprojective algebra $\Pi$ associated to $(C,D)$ and a quiver $\Omega$:

• $\Pi$ is defined as the quotient of the doubled quiver path algebra by specific homogeneous relations reflecting the Cartan data.
• The bigrading structure on $\Pi$ assigns degrees in powers of $q$ and $t$ based on the entries of $C$ and $D$.
• The algebra $\Pi$ is constructed in the sense of Geiß - Leclerc - Schröer for the Langlands dual (transposed Cartan) (Fujita et al., 2021, Fujita et al., 2023).

The entries of the deformed Cartan matrix and their inverses have categorical realization:

• Each $C_{ij}(q,t)$ is recovered from graded dimensions in $\Pi$-modules

$C_{ij}(q,t) = q^{-d_j} t \dim_\Gamma(e_i \bar P_j)$

where $\bar P_j$ is an indecomposable projective with modifications imposed by the algebra relations (Fujita et al., 2023).

• Combinatorially, coefficients in $C(q,t)^{-1}$ enumerate the graded Euler–Poincaré pairings between explicit module classes: $c_{ij}(u,v)$ is (up to sign) the dimension of $\mathrm{Hom}_\Pi(q^{u}t^{v}E_i,S_j)$ for suitable $E_i, S_j$ (Fujita et al., 2021).

4. Representative Examples in Low Rank

In type $A_2$ ($I = \{1,2\}, d_1 = d_2 = 1$)

$$C(q,t) = \begin{pmatrix} q t^{-1} + q^{-1} t & -1 \ -1 & q t^{-1} + q^{-1} t \end{pmatrix}$$

with determinant $\Delta(q,t) = q^2 t^{-2} + 1 + q^{-2} t^2$ and

$$C(q,t)^{-1} = \frac{1}{\Delta(q,t)} \begin{pmatrix} q t^{-1} + q^{-1} t & 1 \ 1 & q t^{-1} + q^{-1} t \end{pmatrix}$$

(Fujita et al., 2021, Fujita et al., 2023).

In type $A_3$ ($d_i = 1$), $C(q,t)$ is tridiagonal with off-diagonal entries $-1$, and $C(q,t)^{-1}$ expressed either by summing a series in $(qt^{-1})$ or via adjugate formula (Fujita et al., 2021).

5. Comparison to Alternative Deformations

The mass-deformed Cartan matrices introduced by Kimura–Pestun for quiver $\mathcal{W}$-algebras feature extra parameters $(q_1, q_2, \mu_e)$. Mapping these via

$$q_1 \mapsto q^2,\quad q_2 \mapsto t^{-2},\quad \mu_e \mapsto q^{d_{ij}} t^{-1} \mu_{ij}^{(g)}$$

aligns their mass-deformed matrix $_{ij}$ with $q^{-D} t C(q, t, \{\mu\})$ precisely when $f_{ij}=1$ or $f_{ji}=1$ for all adjacent $i, j$ (Fujita et al., 2023). This equivalence holds for all symmetric, finite, and untwisted-affine types, ensuring the two deformation frameworks coincide in these cases.

6. Homological and Langlands Dual Structures

The categorical context is strengthened by:

• Braid-group generators categorified via tensoring with ideals $J_i = \Pi(1-e_i)\Pi$, acting on the Grothendieck group as the algebraic $T_i$.
• Periodic projective resolutions for key modules (generalized simples $E_i$, injectives $I_i$), whose kernels and Euler–Poincaré pairings encode the periodicity and duality reflecting the Cartan structure and its deformation (Fujita et al., 2021).
• Switching from $(C, D)$ to $({}^t C, D^{-1} r)$ interchanges the gradings $(q \leftrightarrow t)$ and matches quantum affine $U'_q(\widehat{g})$ with its Langlands dual $U'_t({}^L\widehat{g})$, in line with deformed $\mathcal{W}_{q,t}$-algebras.

7. Applications and Conjectural Correspondence

The $(q,t)$-deformation is fundamentally linked to monoidal categorification problems for quantum affine algebras:

• Generic kernels $K_k^{(i)}$—periodic self-extensions fitting into exact sequences—yield first extension groups $\mathrm{Ext}^1(K_k^{(i)}, K_l^{(j)})$ calculated via the coefficients $c_{ij}(u)$ in $C(q,1)^{-1}$ (Fujita et al., 2021).
• The central conjecture posits a correspondence between the pole-orders of normalized $R$-matrix denominators for Kirillov–Reshetikhin modules and the graded dimensions of $\mathrm{Ext}^1$ spaces:

$$\mathfrak{o}(V_k^{(i)}, V_l^{(j)}) = \dim_q \mathrm{Ext}^1(K_k^{(i)}, K_l^{(j)})$$

validated across non-simply-laced and exceptional types (Fujita et al., 2021).

• This correspondence indicates a deep connection between monoidal and additive-categorical structures in quantum algebra.

The theory of $(q,t)$-deformed Cartan matrices and their categorical avatars provides a unified framework for grading, duality, and extension phenomena in quantum groups, quiver varieties, and their monoidal categorifications (Fujita et al., 2021, Fujita et al., 2023).