Superquivers: Graded Combinatorial & Algebraic Structures

• Superquivers are Z₂-graded extensions of classical quivers, incorporating parity and sign rules to model supersymmetric algebra structures.
• They underpin the construction of quiver Hecke superalgebras and their Clifford counterparts, enforcing parity conditions in polynomial and quadratic relations.
• Superquivers encode Hom-Ext relations in derived categories, facilitating categorification in quantum groups and applications in supersymmetric gauge theories.

A superquiver is a combinatorial and algebraic object that generalizes the classical notion of a quiver by equipping its structure with additional “super” data, typically a $\mathbb{Z}_2$-grading, parity assignments, or extra sign rules. Such enhancements reflect the requirements of supersymmetric representation theory and algebra, and are central to the categorification of various structures associated with quantum groups, Kac–Moody algebras, and derived categories. Superquivers support the algebraic definition and paper of superalgebras, particularly those arising as “super” analogues of Khovanov–Lauda–Rouquier (KLR) algebras, Hecke–Clifford algebras, Hom-Ext structures in the context of derived categories, and categorified invariants in geometry and physics.

1. Foundational Structure: Parity and Grading in Superquivers

The key augmentation that distinguishes a superquiver from a standard quiver is the presence of a parity function, typically a map

$\operatorname{par} : Q_0 \rightarrow \mathbb{Z}_2$

assigning to each vertex either even (0) or odd (1) parity. In certain constructions, arrows may also carry a degree or parity assignment. This grading induces supercommutation rules in the associated (super)path algebra: for instance, polynomial generators associated to odd vertices commute up to a sign:

$$zw = (-1)^{\operatorname{par}(i)\operatorname{par}(j)} wz$$

for variables $z, w$ attached to vertices $i, j$ (see [(Kang et al., 2011), Definition 3.1]).

This refinement percolates through every presentation of the superquiver’s algebra—generator relations, symmetric polynomials, and the form of defining potentials. In many constructions, such as the quiver Hecke superalgebras ($R_n$), a crucial role is played by the requirement that skew-polynomial relations and quadratic constraints respect these parity distinctions, compelling, for odd vertices, that only even powers or combinations appear centrally. In path algebra formulations, as in the Hom-Ext superquiver case, the path algebra is $\mathbb{Z}_2$-graded and composition of degree 1 arrows (representing extensions) is square-zero:

$$\text{If } \deg(\alpha_1) + \deg(\alpha_2) \geq 2,\quad \alpha_1\alpha_2 = 0$$

(as in (Igusa et al., 19 Sep 2025)).

2. Algebraic Realizations: Quiver Hecke Superalgebras and Their Clifford Counterparts

Superquivers form the backbone of several classes of superalgebras:

Quiver Hecke Superalgebras ($R_n$)

These are generated by polynomial (even) variables $x_p$, braid-like operators $T_a$, and idempotents $e(\nu)$, with relations twisted by the vertex parities:

• Quadratic relation:

$$T_a^2 e(\nu) = Q_{\nu_a,\nu_{a+1}}(x_a,x_{a+1})e(\nu)$$

where $Q_{i,j}(w,z)=Q_{j,i}(z,w)$ and $Q_{i,j}(w,z) = Q_{i,j}(-w,z)$ for $i$ odd.

Quiver Hecke–Clifford Superalgebras ($RC_n$)

Arising by “Cliffordizing” the above, they introduce odd Clifford generators $C_p$ satisfying

$$C_p^2=1, \quad C_p C_q + C_q C_p = 0 \quad (p \neq q)$$

and further relations intertwining $T_a, C_p$, and idempotents. The family $RC_n$ is defined over an index set $J$ with an involution $c$ and is designed to admit an equivalence (in the sense of weak Morita superequivalence) with $R_n$, up to Clifford twist and block decomposition (see [(Kang et al., 2011), §2.4, Theorem 3.13]).

Such structures connect, after suitable completions, with affine Hecke–Clifford and affine Sergeev superalgebras, yielding isomorphisms at the level of completed algebras (Theorems 4.4 and 5.4 in (Kang et al., 2011)).

3. Superquivers in Derived and Triangulated Categories: The Hom-Ext Quiver Generalization

In the context of exceptional collections and derived categories, superquivers arise as a categorical encoding of Hom and Ext relations. The Hom-Ext superquiver [Editor’s term] has:

• Vertices: elements of the exceptional collection,
• Arrows: assigned degree $0$ for irreducible morphisms (Hom), degree $1$ for extensions (Ext$^1$),
• Composition: Only mixed compositions (zero + one) can be nonzero, while any path of degree $\geq 2$ is zero,

$$\text{If for a path } p = \alpha_1\cdots\alpha_n,\quad \deg(p)=\sum_i \deg(\alpha_i)\geq 2,\quad p = 0$$

(as in (Igusa et al., 19 Sep 2025)).

A representation assigns, for vertex $v$, an object $M_v$ in a triangulated category, and to each degree $0$ (resp. 1) arrow, a morphism $M_\alpha: M_v \to M_w$ (resp. $M_\beta: M_v \to \Sigma M_w$), modeling morphisms and extension classes.

In certain settings (e.g., derived categories of hereditary algebras), the algebraic structure of the superquiver fully encodes the exceptionality, with twist functors and Dehn twists acting naturally on the superquiver, which becomes a robust invariant under derived autoequivalence in type $\widetilde{\mathbb{A}}$ (Igusa et al., 19 Sep 2025).

4. Applications to Categorification and Representation Theory

Superquivers are essential in the categorification of quantum groups, especially in the super setting:

• In Kac–Moody and quantum group theory, quiver Hecke (super)algebras $R_n$ categorify the negative half of quantum enveloping algebras, with parity data necessary for categorifying Lie superalgebras and their modules—significantly extending the canonical basis and crystal combinatorics familiar from purely even cases (Kang et al., 2011).
• The notion of weak Morita superequivalence between $R_n$ and $RC_n$ is central, ensuring their module categories (possibly as direct sums with Clifford twist) become equivalent as supercategories, a critical step for categorification programs.
• In representation theory of algebras, superquivers serve as organizing principles for the module categories over superalgebras, encoding parity and extension data in combinatorial form. In derived categories, superquivers encapsulate both morphism and extension information in a minimal and invariant way under derived and geometric autoequivalences.

5. Mathematical Formulations and Key Structural Properties

Some of the key formulas and definitions appearing in superquiver theory include:

• Vertex parity splitting: $I = I_{\text{even}} \sqcup I_{\text{odd}}$, $\operatorname{par}(i) \in \{0,1\}$.
• Twisted commutation: For variables $w, z$ of parity $\operatorname{par}(i), \operatorname{par}(j)$, $$zw = (-1)^{\operatorname{par}(i)\operatorname{par}(j)}wz$$.
• Quadratic and symmetry relations in the superalgebra:

$$Q_{i,j}(w,z) = Q_{j,i}(z,w),\qquad Q_{i,j}(w,z) = Q_{i,j}(-w,z) \quad (i \text{ odd})$$

• Clifford relations for generators $C_p$:

$C_p^2 = 1,\quad C_pC_q + C_qC_p = 0$

• Weak Morita superequivalence:

$$\text{If } A = A_1 \oplus A_2,\quad B = B_1 \oplus B_2,\quad \mathrm{Mod}(A_1)\simeq\mathrm{Mod}(B_1),\ \mathrm{Mod}(A_2)\simeq \mathrm{Mod}(B_2)^c$$

These properties enforce a strong rigidity on the definition and deformation of superquivers and their associated (super)algebras.

6. Superquivers in Physics and Topological Invariants

Superquivers are intrinsic to the modeling of protected spectra and BPS states in supersymmetric gauge theories:

• In 4d $\mathcal{N}=2$ gauge theory, quivers (sometimes with superpotential) are used to model BPS spectra, with superquivers naturally incorporating parity data relevant for spin and fermionic extensions (Cecotti, 2012).
• In knot homology and the knots-quivers correspondence, superquiver structures are pivotal for expressing the generating functions for colored HOMFLY-PT and superpolynomials in a universal quiver (symmetric) form that matches motivic Donaldson–Thomas invariants. The integral structure and block structure of the quiver matrix $C$ directly reflect the bigraded nature of knot homologies (Kucharski et al., 2017).
• In derived and triangulated categories, superquivers facilitate the paper of exceptional collections and their mutations or autoequivalences, underpinning geometric models such as those using arc diagrams and Dehn twist actions (Igusa et al., 19 Sep 2025).

7. Classification, Universality, and Future Directions

The classification of superquivers remains intertwined with:

• The existence of associated (super)algebras with prescribed growth conditions or representation type,
• The encoding of parity and extension data in highly structured combinatorial objects,
• The extension to universal objects: universal quivers can in principle be “superized” to support any parity-decorated full subquiver, which would categorize all possible superquivers up to mutation equivalence (Fomin et al., 2020).

Further research is prompted on the systematic paper of superquivers as invariants in representation theory, their interplay with cluster theory, and the development of categorical and geometric models that capture the enhanced symmetries and dualities peculiar to superalgebraic settings.

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Superquivers thus provide a precise formalism for encoding and analyzing $\mathbb{Z}_2$-graded data in quiver-theoretic frameworks, supporting significant advances in algebraic categorification, representation theory, derived category invariants, and the algebraic structures underlying supersymmetric quantum field theories.