Isomeric Kac–Moody Categorification

• Isomeric Kac–Moody categorification is a 2-categorical framework that generalizes classical approaches by incorporating parity and Clifford tokens to model Q-type representations.
• It builds on the isomeric Heisenberg supercategory, employing diagrammatic calculus with odd bubbles, dot operators, and specialized crossings to encode spectral theory.
• The construction bridges classical and super Kac–Moody theories, offering fresh insights into quantum covering algebras through Clifford twists and refined Cartan data.

Isomeric Kac–Moody categorification is the 2-categorical framework underlying the “Q-type” or isomeric super analogues of Kac–Moody representation theory, especially as it relates to the supergroup Q(n) and covering (super)algebras. This subject provides a bridge between Heisenberg categorification with super and Clifford symmetry (the “isomeric” Heisenberg approach) and fully-fledged Kac–Moody 2-categories encoding odd, super, or covering phenomena. It refines and generalizes classical Kac–Moody categorification by incorporating parity, Clifford tokens, and more general Cartan data, capturing the categorical structure of representations relevant for supergroups and quantum covering algebras (Brundan et al., 23 Nov 2025, Ellis et al., 2013).

1. Isomeric Heisenberg Supercategory and Its Structure

The isomeric Heisenberg supercategory $\mathrm{Heis}_\kappa(\mathcal{C})$, as established by Brundan and Savage, encodes the categorical version of the Heisenberg algebra with super and Clifford features. The generating objects are $P$ (arrow-up) and $Q$ (arrow-down), with 1-morphisms being arbitrary tensor powers. Notably, the 2-morphisms are generated by:

• Clifford-token $c: P \to P$, parity 1, with $c^2 = -\mathrm{id}$,
• Dot $x: P \to P$, parity 0,
• Crossings $P \otimes P \to P \otimes P$ (even),
• Cups $\eta: 1 \to Q \otimes P$, and caps $\epsilon: P \otimes Q \to 1$ (even),
• Composite and tensor products of these, subject to affine Sergeev (super) and inversion/curl relations.

Characteristic relations include the zig-zag adjunctions (expressing $P \dashv Q$), odd-bubble vanishing, and the affine Sergeev relations, as summarized in the table:

Generator	Symbol	Parity	Key Relation(s)
Clifford token	$c$	1	$c^2 = -\mathrm{id}$
Dot	$x$	0	$cx = -xc$
Crossing	—	0	Usual swap + super sign rules
Cup/Cap	$\eta, \epsilon$	0	Zig-zag: $P \dashv Q$

The explicit inclusion of a Clifford-token is the fundamental “isomeric” feature, necessitating a super- or Clifford-categorical structure and dictating a richer spectral theory for dot operators (Brundan et al., 23 Nov 2025).

2. Construction of the Isomeric Kac–Moody 2-Category

The isomeric Kac–Moody 2-category, denoted $V(\mathfrak{g})$, is a strict 2-supercategory determined by a “super Cartan datum,” generalizing classical Kac–Moody data to types B, C, and twisted A using parity assignments on simple roots. The main features are:

• Objects: Weights $\lambda$ indexed by a lattice $X$.
• 1-morphisms: For each $i$ in the index set $I$, $P_i 1_\lambda: \lambda \to \lambda + \alpha_i$ and $Q_i 1_\lambda: \lambda \to \lambda - \alpha_i$.
• 2-morphisms: Even dots (on all strands), odd Clifford-tokens (on odd-index strands), even crossings (with power-series coefficients $g_{ij}(x_i,y_j)$), as well as cups/caps and invertible “sideways” crossings.

Defining relations include:

• Zig-zag adjunctions ($P_i \dashv Q_i$),
• Quiver Hecke–Clifford superalgebra relations (Clifford, dot, and crossing relations),
• Inversion and odd-bubble vanishing (forcing certain diagrammatic nilpotency).

This structure directly categorifies isomeric (covering) Kac–Moody quantum algebras, generalizing the Khovanov–Lauda–Rouquier framework to Q-type settings (Brundan et al., 23 Nov 2025).

3. The Passage: Spectral Functor and the Induction Mechanism

A critical achievement is the construction of a spectral 2-functor $$\Psi: V(\mathfrak{g}) \to \mathrm{End}(\mathcal{R})$$, where $\mathcal{R}$ is an Abelian supercategory with a strict action of the isomeric Heisenberg supercategory. The essential mechanism is as follows:

• The dot $x: P \to P$ has a minimal polynomial $m_V(x)$ determined by the Clifford structure.
• $P$ and $Q$ decompose into generalized eigenspaces $P_i$, $Q_i$ indexed by $i \in I$.
• The subcategories $\mathcal{R}_\lambda$ partition objects according to weight, coordinating with $P_i, Q_i$ as functors shifting weights.
• The structural power series $g_{ij}(x_i,y_j)$, $h_i(x_i,y_i)$, etc., are used to define the action of crossings and higher morphisms, encoding the “quiver Hecke–Clifford” relations.
• Verifying explicitly that all (A)–(E) relations of $V(\mathfrak{g})$ hold, one establishes a strict 2-representation.

This functorial bridge induces a categorification sequence: isomeric Heisenberg $\rightarrow$ isomeric Kac–Moody $\rightarrow$ super Kac–Moody (upon further “Clifford twisting”) (Brundan et al., 23 Nov 2025).

4. Special Cases and Examples

(a) Rank 1 Case: Odd $\mathfrak{sl}_2$ and the Affine Brauer–Clifford Category

The rank 1 instance (with $I = \mathbb{Z}$) and central charge $\kappa=0$ recovers the degenerate affine Brauer–Clifford category. The dot and crossing coefficients specialize to $x_0 = x$, $x_i = x - i$, with $g_{ii}(x_i, y_i) = 1/(x_i - y_i)$, matching the odd quiver Hecke relations for $\mathfrak{sl}_2$. The resulting diagrammatic calculus and odd nilHecke structure coincide with the “odd” categorifications of $U_q(\mathfrak{sl}_2)$ developed by Ellis–Lauda, with $\pi = \pm 1$ selecting between the odd and super forms (Ellis et al., 2013).

(b) Rank 2 and Beyond

For $I = \{\dots, -1, 0, 1, 2, \dots\}$ and Cartan of type B/C, the interaction between even and odd nodes is mediated by specific power-series solutions ($g_{01}, t_i$) to the quiver Hecke–Clifford relations. The existence of nontrivial invertibility conditions (e.g., the invertibility of certain crossing matrices) is demonstrated explicitly (Brundan et al., 23 Nov 2025).

5. Specializations, Decategorification, and Grothendieck Theory

Grothendieck groups of isomeric Kac–Moody 2-categories are modules over suitable group algebras reflecting $\mathbb{Z} \times \mathbb{Z}_2$ gradings. Specialization of parameters (notably $\pi$ in the covering quantum case) links isomeric structures to their classical or super counterparts:

• $\pi = +1$: Recovers Lusztig/Launda’s odd $\mathfrak{sl}_2$ categorification; the covering relation reduces to the classical quantum group relation, and the odd nilHecke replaces the even nilHecke (Ellis et al., 2013).
• $\pi = -1$: Yields the supercategorification of $\mathfrak{osp}(1|2)$; relations become supercommutative by parity, with diagrammatic signs dictated by odd bubbles and parity interchange (Ellis et al., 2013).

These mechanisms underpin the correspondence between the diagrammatic, parity-enriched frameworks of superrepresentation theory and the algebraic structures of quantum covering and superalgebras.

6. Relation to Classical and Super Kac–Moody Categorification

Classically, Heisenberg categorification for type A (GL-type) passes to ordinary Kac–Moody 2-categories (e.g., Rouquier’s $U(\mathfrak{g})$) via evaluation at spectrum of the dot endomorphism. In the isomeric (Q-type) case, the Clifford-token and enforced nilpotency alter spectral theory, requiring a two-step process:

1. Heis$_\kappa(\mathcal{C})$ to isomeric Kac–Moody: Decompose generators using Clifford spectral theory and match quiver Hecke–Clifford relations.
2. Isomeric Kac–Moody to super Kac–Moody: Apply a further Clifford twist (to be addressed in future work), reconciling with the Kang-Kashiwara-Tsuchioka supercategorification schemes.

This hierarchy situates the isomeric theory as a categorification of phenomena intrinsic to Q-type supergroups and Clifford-symmetric representation categories—a significant generalization beyond the classical and even super cases (Brundan et al., 23 Nov 2025).

7. Diagrammatics, Graded Superstructure, and Future Directions

The diagrammatic calculus incorporates parity (dotted, dashed strands, parity shifts), Clifford tokens, odd-bubble relations, and power-series crossings. The defining relations ensure that all diagrammatic morphisms respect super and Clifford symmetry, accommodating the extra sign conventions and nilpotent/Clifford structure necessary for Q-type categorification.

Current and future developments include:

• Further Clifford twists to relate isomeric and super Kac–Moody categorifications,
• Classification of Abelian and derived 2-representations for broader root systems,
• Extensions to the categorification of covering quantum groups and quantum symmetric pairs,
• Applications to link homologies and higher representation theory, particularly for supergroups with pronounced Q-type or Clifford symmetry (Brundan et al., 23 Nov 2025).

These advancements confirm the foundational role of isomeric Kac–Moody categorification in the modern representation theory of superalgebras and quantum groups.