Kac-Moody 2-Category

• Kac–Moody 2-categories are graded, k-linear 2-categories that categorify the enveloping algebra U(g) using generators and relations inspired by nil affine Hecke algebras.
• The construction integrates categorical Chevalley generators, adjunction maps, and diagrammatic 2-morphisms (dots and crossings) to mirror core algebraic structures.
• Decategorification via the Grothendieck group K0 recovers the integral form of U(g), ensuring control over integrable 2-representations and applications in geometric and derived settings.

A Kac-Moody 2-category, as introduced by Rouquier, is a $k$-linear, graded 2-category $\mathfrak{A}(\mathfrak{g})$ that categorifies the enveloping algebra $U(\mathfrak{g})$ (or its quantum analogue) for a given symmetrizable Kac–Moody algebra $\mathfrak{g}$. The decategorification of $\mathfrak{A}(\mathfrak{g})$ via the split Grothendieck group $K_0$ recovers the integral form (or positive/negative parts) of $U(\mathfrak{g})$. The construction is by generators and relations, generalizing the structure of the nil affine Hecke algebra and the Chuang–Rouquier categorical $\mathfrak{sl}_2$-theory to arbitrary type.

1. Generators and Defining Data

The objects of $\mathfrak{A}(\mathfrak{g})$ are the weights $\lambda$ in the weight lattice $X$. For each simple root $s$ in the indexing set $I$, the 1-morphisms are generated by the categorified Chevalley generators:

• $E_s : \lambda \to \lambda + \alpha_s$, and
• $F_s : \lambda \to \lambda - \alpha_s$,

where $F_s$ is both left and right adjoint (up to canonical isomorphism) to $E_s$.

The 2-morphisms consist of diagrammatic data:

• Dots $x_s \in \operatorname{End}(E_s)$,
• Crossing morphisms $T_{st}: E_s E_t \to E_t E_s$ (for all $s,t \in I$),
• Adjunction maps $\eta_s : \operatorname{Id} \to F_s E_s$ and $\varepsilon_s : E_s F_s \to \operatorname{Id}$.

These are subject to extensive relations mirroring and categorifying the relations of $U(\mathfrak{g})$.

2. Relations: Nil Affine Hecke Algebra and Adjunctions

The core relations among the 2-morphisms are modeled on the (degenerate) nil affine Hecke algebra:

• Bubble/crossing relations: For $s, t \in I$,

$T_{st} T_{ts} = Q_{st}(E_t x_s, x_t E_s),$

where $Q_{st}(u,v)$ is a Cartan datum-dependent polynomial arising from the quiver Hecke algebra context.

• Dot–crossing commutation:

$$T_{st} \circ (x_s E_t) - (E_s x_t) \circ T_{st} = \delta_{st}(\text{up to sign}),$$

categorifying the corresponding degenerate affine Hecke algebra relation.

• Adjunction/inversion relations: The adjunction data yield isomorphisms

$$E_s F_s \oplus (\text{shifted copies of the identity}) \cong F_s E_s \oplus (\text{other shifted copies}),$$

reflecting the commutator $[e_s, f_s] = h_s$ at the decategorified level.

The cyclicity, infinite Grassmannian (bubble) relations, and more, ensure the correspondence with symmetric functions and Hecke-type induction/restriction on tensor powers $E_s^n$.

3. Grading and Decategorification

$\mathfrak{A}(\mathfrak{g})$ admits a grading compatible with the Cartan matrix:

• $\deg(x_s) = 2d_s$, $\deg(T_{st}) = -b_{st}$ for $d_s$, $b_{st}$ fixed by symmetrization/data.
• The Grothendieck group $K_0(\mathfrak{A}(\mathfrak{g}))$ canonically recovers $U_+(\mathfrak{g})$ (or the integral form), i.e.

$$K_0(\mathfrak{A}(\mathfrak{g})) \cong U_+(\mathfrak{g}),$$

and, in 2-representations, $K_0$ functorially realizes the associated $U(\mathfrak{g})$-module.

This means $K_0$-level properties can be used for “control”—verifying that 2-categorical relations hold by checking the $K_0$-decategorified image.

4. 2-Representations

Given a $k$-linear 2-category $\mathcal{C}$ (e.g., abelian, triangulated, or module categories), a 2-representation is a strict 2-functor

$$R: \mathfrak{A}(\mathfrak{g}) \longrightarrow \mathcal{C}.$$

The most natural/tractable class comprises integrable 2-representations: the functors $E_s$, $F_s$ act locally nilpotently. This ensures the finite nature of relevant weight spaces, is critical for highest-weight theory, and guarantees the module structure on $K_0$ reflects integrable $U(\mathfrak{g})$-representation theory.

Morphisms of 2-representations, as well as their structure (Jordan–Hölder-type, blocks, etc.), are studied via adjunction data, fully-faithfulness, and passage to $K_0$. Checking invertibility or isomorphism of certain 2-morphisms often reduces to $K_0$-level arguments.

5. Construction by Generators and Relations: Role of Nil Affine Hecke Algebra

The essential innovation is the definition of the 2-category by generators and relations paralleling nilHecke-induced structures:

• For type A ($\mathfrak{sl}_n$), the endomorphism algebras $E_s^n$ are isomorphic to degenerate/affine Hecke algebras, so the combinatorics of symmetric groups and Hecke algebra modules directly control categorical induction/restriction (e.g., $End(E_s^n) \cong n!\cdot k[x_1,\dots,x_n]$).
• More generally, for arbitrary Cartan data, one uses quiver (nil) Hecke algebras governed by relations encoding $(Q_{st})_{s,t \in I}$, with

$T_{st} T_{ts} = Q_{st}(x_s, x_t).$

This ensures the categorified Serre relations are encoded in the 2-category, e.g., higher commutators categorify the homotopic vanishing of certain complexes.

6. Connection to Earlier Type A (Chuang–Rouquier) Categorifications

Rouquier’s construction extends Chuang–Rouquier's $\mathfrak{sl}_2$-categorification to general Kac–Moody type:

• In type A, the combinatorics of the nil affine Hecke algebras link the categorical structures to cyclotomic Hecke algebras and symmetric group representation theory, particularly in the explicit block description of module categories.
• For general $A$, the nilHecke/quiver Hecke framework defines the necessary relations and ensures that, under decategorification, the integral/graded categorifications correspond to the correct universal enveloping algebra structures.
• The construction allows for specializations (e.g., to quiver representations, or upon ring specializations) resulting in integral or graded 2-categories.

7. Applications and Structural Impact

The Kac-Moody 2-category $\mathfrak{A}(\mathfrak{g})$ serves as a universal object in higher representation theory:

• Its 2-representations systematically categorify integrable $U(\mathfrak{g})$-modules and functorial structures (including induction, restriction, and dualities).
• The presence of dual adjoints and explicit isomorphism decompositions provides a “categorified” commutator theory, crucial for modeling categorical braid group actions and understanding derived, geometric, and homotopical symmetries.
• The construction generalizes and envelopes the full representation-theoretic content of both finite and affine type via a universal 2-categorical calculus.

This approach has become a cornerstone of higher representation theory, including the link to categorification of quantum groups, geometric representation theory, categorified braid group actions, and derived equivalences. The heavy reliance on $K_0$ as a control mechanism means that deep structural results about $U(\mathfrak{g})$-modules are reflected in and often determined by their higher-categorical avatars.

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Key Data and Relations Table

Element	Notation	Role
Objects	$\lambda \in X$	Weight lattice elements
Generating 1-morphisms	$E_s$, $F_s$	Categorical Chevalley generators
Dot 2-morphisms	$x_s \in \mathrm{End}(E_s)$	Categorical multiplication by polynomials
Crossing 2-morphisms	$T_{st}: E_s E_t \to E_t E_s$	Braid/Hecke-type relations
Adjunctions	$\eta_s, \varepsilon_s$	Categorical (co)commutators
Grading	$\deg x_s = 2d_s$, $\deg T_{st} = -b_{st}$	Compatibility with Cartan data
Decategorification	$$K_0(\mathfrak{A}(\mathfrak{g})) \cong U_+(\mathfrak{g})$$	Recovers enveloping algebra

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Summary

Rouquier's Kac–Moody 2-category construction (0812.5023) provides a generator-and-relation presentation for a graded, $k$-linear 2-category whose decategorification recovers the structure of the (integral form of the) enveloping algebra $U(\mathfrak{g})$. Its 2-representations categorify modules, with integrability and control over higher-categorical relations ensured via $K_0$. The construction unifies and generalizes earlier type A and $\mathfrak{sl}_2$-categorifications and underpins modern approaches to categorification of quantum groups, braid group actions, and geometric representation theory.