[Paper Review] Rigidity in higher representation theory

Sabin Cautis|arXiv (Cornell University)|Sep 2, 2014

Algebraic structures and combinatorial models22 references17 citations

TL;DR

This paper introduces a $(\mathfrak{g},\theta)$ action as a minimal framework for categorical $\mathfrak{g}$-actions, proving that such actions naturally carry an action of quiver Hecke algebras (KLR algebras) via homological computations. The key result is a rigidity phenomenon: under mild conditions, the endomorphism algebras of compositions of $\mathfrak{g}$-functors are determined by the Lie algebra $\mathfrak{g}$, generalizing Khovanov-Lauda categorification to naive categorical actions.

ABSTRACT

We describe a categorical g action, called a (g,theta) action, which is easier to check in practice. Most categorical g actions can be shown to be of this form. The main result is that a (g,theta) action carries actions of quiver Hecke algebras (KLR algebras). We discuss applications of this fact to categorical vertex operators, affine Grassmannians (or Nakajima quiver varieties) and to homological knot invariants.

Motivation & Objective

• To establish a minimal framework for categorical $\mathfrak{g}$-actions that is easier to verify in examples than full Khovanov-Lauda-Rouquier 2-representations.
• To resolve the problem that naive categorical $\mathfrak{g}$-actions do not a priori carry quiver Hecke algebra actions, which are essential for categorified quantum groups.
• To prove that under a $(\mathfrak{g},\theta)$-action structure, the endomorphism algebras of compositions of $\mathfrak{g}$-functors are governed by KLR algebras, thus establishing rigidity.
• To extend the applicability of Khovanov-Lauda 2-representation theory to broader classes of categorical $\mathfrak{g}$-actions by identifying sufficient conditions for KLR action.

Proposed method

• Introduces the concept of a $(\mathfrak{g},\theta)$ action as a minimal categorical $\mathfrak{g}$-action with additional data $\theta$ to control homological constraints.
• Performs a sequence of $\operatorname{Hom}$-space calculations using adjunction and isomorphisms from the categorical $\mathfrak{g}$-action axioms to constrain natural transformations.
• Constructs the $T_{ij}$, $X_i$, and $T_{ijk}$ maps via dimension-counting and nonvanishing arguments in $\operatorname{Hom}$-spaces between compositions of $\mathfrak{g}$-functors.
• Establishes the affine nilHecke algebra action by verifying the required relations (e.g., $T_{iji} = T_{jij}$, $T_{ijk} = T_{jik}$) through homological computations.
• Uses transient maps to handle negligible 2-morphisms, with the condition that modding out by transients is unnecessary when $\mathfrak{g} = \mathfrak{sl}_n$.
• Applies results from [CLa] to deduce that a $(\mathfrak{g},\theta)$ action induces a 2-representation in the Khovanov-Lauda sense, modulo transient maps.

Experimental results

Research questions

• RQ1 Can a naive categorical $\mathfrak{g}$-action—without assumed divided powers or Serre relations—still carry an action of the quiver Hecke algebra (KLR algebra)?
• RQ2 What minimal additional structure ($\theta$) is needed to ensure that the endomorphism algebras of compositions of $\mathfrak{g}$-functors are governed by KLR algebras?
• RQ3 To what extent is the structure of natural transformations between compositions of $\mathfrak{g}$-functors rigidly determined by the Lie algebra $\mathfrak{g}$ alone?
• RQ4 Under what conditions can transient (negligible) 2-morphisms be safely ignored in the construction of KLR algebra actions?
• RQ5 Does the rigidity result extend to all Kac-Moody algebras, or are there obstructions beyond $\mathfrak{sl}_n$?

Key findings

• A $(\mathfrak{g},\theta)$ action carries an action of the quiver Hecke algebra modulo transient maps, as stated in Theorem 2.2.
• The endomorphism algebra $\operatorname{End}({\sf{E}}_i{\sf{E}}_i)$ is constrained by the $\mathfrak{g}$-action data, enabling the construction of $X_i$ and $T_{ii}$ maps.
• The dimension of $\operatorname{Hom}({\sf{E}}_i{\sf{E}}_j{\sf{E}}_k{\mathbf{1}}_\lambda, {\sf{E}}_k{\sf{E}}_j{\sf{E}}_i{\mathbf{1}}_\lambda\langle 3\rangle)$ is at most 1, with equality if and only if certain weight projectors ${\mathbf{1}}_\mu$ are nonzero.
• For distinct $i,j,k$, $\dim \operatorname{Hom}({\sf{F}}_k{\sf{E}}_i{\sf{E}}_j{\mathbf{1}}_\lambda, {\sf{E}}_j{\sf{E}}_i{\sf{F}}_k{\mathbf{1}}_\lambda\langle -\langle i,j\rangle \rangle) = 1$ if and only if all relevant weight projectors are nonzero.
• When $\mathfrak{g} = \mathfrak{sl}_n$, the transient map condition can be omitted, and the KLR algebra action holds without quotienting.
• The result implies that a $(\mathfrak{g},\theta)$ action induces a 2-representation in the Khovanov-Lauda sense, modulo transient maps.