Isomeric Heisenberg Categorification

• Isomeric Heisenberg categorification is a framework that constructs categorical analogues of the Heisenberg algebra for queer supergroups using Clifford superalgebras.
• It adapts classical type A categorification by introducing new generators, relations, and combinatorial techniques for spin symmetric and super settings.
• This approach underpins isomeric Kac–Moody categorification, linking diagrammatic methods with advanced representation theory of Q(n) and related structures.

Isomeric Heisenberg categorification is the process of constructing categorical analogues of the Heisenberg algebra and its modules that are adapted to the $Q$-type setting, specifically the context of the queer supergroup $Q(n)$, spin symmetric groups, and category $\mathcal{O}$ for $\mathfrak{q}_n$. This theory extends classical (type $A$) Heisenberg categorification frameworks—central tools for controlling the representation theory of symmetric groups, general linear groups, and related structures—to the super and spin settings, with new generators, relations, and combinatorics reflecting the role of Clifford superalgebras and the affine Sergeev or Brauer–Clifford superalgebras. The construction provides a foundation for isomeric Kac–Moody categorification and ultimately enables categorification of key algebraic structures in $Q$-type representation theory (Brundan et al., 23 Nov 2025).

1. Foundational Motivation and Framework

Traditional Heisenberg and Kac–Moody categorifications, as developed by Khovanov, Mackaay–Savage, Brundan, Webster–Williamson, and others, give monoidal categories acting on Abelian categories; upon decategorification, these yield Heisenberg algebras of integral central charge. These frameworks fit the "type $A$" world, controlling representation theory for $GL$-type objects such as symmetric groups, cyclotomic Hecke algebras, quantum groups of type $A$, and category $\mathcal{O}$ for $\mathfrak{gl}_n$.

Isomeric categorification addresses $Q$-type representation theory, including spin symmetric groups, category $\mathcal{O}$ for $\mathfrak{q}_n$, and representations of $Q(n)$. This domain features distinct Cartan data: types $A_\infty, B_\infty, C_\infty$ in characteristic 0, $A_{p-1}^{(1)}, A_{p-1}^{(2)}$ in positive characteristic, and crucially requires working in the super-setting where the Cartan datum contains an odd simple root at label 0. In place of standard Heisenberg categories, the isomeric setting employs categories whose endomorphism algebras are affine Sergeev or affine oriented Brauer–Clifford superalgebras.

2. Structure of the Isomeric Heisenberg Category

For fixed algebraically closed ground field $\Bbbk$ (char $\ne 2$), central charge $\kappa\in\mathbb{Z}$, and the rank-one Clifford superalgebra $C=\langle c\mid c^2=-1\rangle$ (with $c$ odd and even trace form), the isomeric Heisenberg category $Heis_\kappa(C)$ is defined as a strict monoidal supercategory generated by the following:

• Objects: $P$ ("creation", upward arrow), $Q$ ("annihilation", downward arrow).
• Generating 2-morphisms (even except where noted):
  • Clifford token (odd) on $P$.
  • Dot (even) on $P$.
  • Crossing on $P\otimes P$.
  • Cup and cap giving $Q$ as right (and left) dual to $P$.

The relations are as follows:

1. Zig–zag (adjunction): $Q$ is both left and right dual to $P$.
2. Affine Sergeev superalgebra relations:
   • Braid and idempotent relations for crossings.
   • Clifford token squares to $-1$ ($\bullet\bullet=-\operatorname{id}$), tokens on the same strand anticommute.
   • Dots and tokens obey mixed (anti)commutativity, e.g., $\bullet\circ = -\circ\bullet$ on one strand.
   • Standard dot–crossing relations.
3. Inversion relation: The infinite matrix $M_\kappa$ (whose entries involve crossings, cups, and dotted cups) must be invertible in the additive envelope, ensuring decategorification recovers $[Q,P]=\kappa$.
4. Odd bubble relation: A single-stranded "figure-eight" (odd bubble) with Clifford token is zero, eliminating unwanted odd bubbles.

Collectively, these specify the isomeric Heisenberg supercategory $Heis_\kappa(C)$.

3. Notion and Realization of Isomeric Heisenberg Categorification

An isomeric Heisenberg categorification of central charge $\kappa$ consists of:

• A locally finite Abelian supercategory $R$.
• A biadjoint pair of exact endofunctors $(P,Q)$.
• Even unit and counit morphisms $\mathbf{1}\rightarrow QP$, $PQ\rightarrow\mathbf{1}$.
• Supernatural transformations corresponding to the dot, token, and crossing generators, such that the $Heis_\kappa(C)$ relations hold in $\operatorname{End}(R)$.

Equivalently, this is a strict monoidal super-functor $$\Psi: Heis_\kappa(C)\rightarrow \operatorname{End}(R)$$, with $R$ generated as a Serre subcategory by the action of $P$ and $Q$ on a finite set of objects with purely even supercenter in each endomorphism algebra.

Decategorification: The Grothendieck group $K_0(R)$ (ignoring parity shift) recovers the ordinary Heisenberg algebra: $[K_0(P), K_0(Q)] = \kappa$.

4. Spectral and Weight-Space Decomposition

Nilpotency of the dot on $P$ enables spectral decomposition:

• $P$ and $Q$ decompose as $P=\bigoplus_{i\in I}P_i$, $Q=\bigoplus_{i\in I}Q_i$, with spectral parameters $I\subset\Bbbk$ (specifically, square roots of $i(i+1)$, shifted).
• The pairs $(P_i, Q_i)$ satisfy analogous relations to $Heis_\kappa(C)$, but focus on the eigenvalue $b(i)$; Clifford token induces $P_i\simeq P_{-i}$ and $Q_i\simeq Q_{-i}$.
• A weight function for irreducible $L\in R$ is constructed from the order of poles/zeros of the bubble generating function

$X_L(u) = \sum_{n\geq 0} \langle \text{counterclockwise $n$-dot bubble on$L$} \rangle u^{-n-1}.$

• The resulting decomposition $R= \bigoplus_{\lambda\in X}R_\lambda$ is indexed by the minimal weight lattice $X= \bigoplus_{i\in I}\mathbb{Z}\varpi_i$; $P_i$ and $Q_i$ induce transitions between these weight subcategories.

5. Comparison with Ordinary Heisenberg Categorification

Key structural differences include:

• The supernature, with Clifford token ($c^2=-1$) introducing nilpotent, anticommuting operations not present in type $A$.
• Bubble slides and the odd bubble relation, simplifying the affine Sergeev algebraic presentation and affecting the calculus of diagrams.
• A change of variable $x\mapsto x_i$ for each $i$-colored strand, and the emergence of rational invariants (e.g., $g_{ij}(x_i, y_j)$) tied to the underlying super-Cartan data.
• Dependence of matrix inversion (for $M_\kappa$) on the parity of $i$ and the presence of bubbles with Clifford tokens, in contrast to the uniform behavior found in type $A$ settings.

6. From Isomeric Heisenberg to Isomeric Kac–Moody Categorification

Building on the isomeric Heisenberg framework, the isomeric Kac–Moody 2-category $V(\mathfrak{g})$ is introduced, reflecting the same super-Cartan datum:

• Objects: weights $\lambda\in X$.
• 1-morphisms: divided power functors ($P_i 1_{\lambda}:\lambda\to\lambda+\alpha_i$, $Q_i 1_{\lambda}:\lambda\to\lambda-\alpha_i$).
• 2-morphisms: dots, tokens, crossings, cups/caps, with quiver Hecke–Clifford relations (as per Kang–Kashiwara–Tsuchioka).

Bridge theorem: Any isomeric Heisenberg categorification $R$ furnishes, after decomposing $P,Q$ and passing to weight subcategories, a 2-representation of $V(\mathfrak{g})$. The combinatorial 2-morphisms $g_{ij}(x_i, y_j), f_i(x_i, y_j), h_i(x_i, y_j)$, derived from the change-of-variable and bubble-slide machinery, provide the required relations for crossings and bubble slides in $V(\mathfrak{g})$.

This realizes a complete categorification of the isomeric Heisenberg and Kac–Moody algebras, mirroring the established classical type $A$ narrative (Brundan et al., 23 Nov 2025).

7. Examples and Applications

Applications of these constructions include:

• Category of finite-dimensional modules over the spin-symmetric (Sergeev) superalgebra.
• Rational $Q(n)$-modules.
• Category $\mathcal{O}$ for $\mathfrak{q}_n(C)$.

In each case, the categorical framework leads to structural understanding of phenomena such as integrable crystals, Rickard equivalences à la Chuang–Rouquier, canonical bases, and higher structures.

As a minimal example, consider $R=C\text{-smod}$, the category of finite-dimensional $C$-supermodules. Here,

• $P= C[x]\otimes-$,
• $Q= \operatorname{Hom}_{C[x]}(C[x], - )$,
• the dot is multiplication by $x$,
• the token is the Clifford generator $c$.

All defining relations of $Heis_0(C)$ are satisfied, and $K_0(R)\cong \mathbb{Z}^2$, manifesting the basic Fock-space representation of the Heisenberg algebra of zero charge.

This framework lays the groundwork for a full $Q$-type analogue of classical type $A$ categorifications, accommodating the additional complexities imposed by the queer supergroup $Q(n)$ and its associated algebraic structures (Brundan et al., 23 Nov 2025).