[Paper Review] Unfurling Khovanov-Lauda-Rouquier algebras

Ben Webster|arXiv (Cornell University)|Mar 21, 2016

Algebraic structures and combinatorial models23 references9 citations

TL;DR

This paper proves the long-standing conjecture that the Khovanov-Lauda-Rouquier (KLR) 2-category U(g) categorifying a symmetrizable Kac-Moody algebra g is non-degenerate, with its Grothendieck group isomorphic to the idempotented quantum group ˙U(g). The authors establish this by deforming categorical representations of g, showing that if the deformed representations have the expected dimension at the generic point, they cannot be smaller at the special point, leveraging upper semicontinuity of dimension under deformation. The key contribution is a general theory of 'unfurling'—a construction that relates categorical actions of g to those of a larger Lie algebra ˜g via spectral deformation, resolving non-degeneracy in full generality for arbitrary Cartan data.

ABSTRACT

In this paper, we study the behavior of categorical actions of a Lie algebra $\mathfrak{g}$ under the deformation of their spectra. We give conditions under which the general point of a family of categorical actions of $\mathfrak{g}$ carry an action of a larger Lie algebra $\mathfrak{ ilde{g}}$, which we call an {\bf unfurling} of $\mathfrak{g}$. This is closely related to the folding of Dynkin diagrams, but to avoid confusion, we think it is better to use a different term. Our motivation for studying this topic is the difficulty of proving that explicitly presented algebras and categories in the theory of higher representation theory have the "expected size." Deformation is a powerful technique for showing this because of the upper semicontinuity of dimension under deformation. In particular, we'll use this to show the non-degeneracy (in the sense of Khovanov-Lauda) of the 2-quantum group $\mathcal U$ for an arbitrary Cartan datum and any homogeneous choice of parameters.

Motivation & Objective

• To resolve the long-standing open problem of proving non-degeneracy of the Khovanov-Lauda-Rouquier 2-category U(g) for arbitrary symmetrizable Kac-Moody algebras g.
• To establish that the Grothendieck group of U(g) is isomorphic to the idempotented universal enveloping algebra ˙U(g), confirming the conjecture of Khovanov, Lauda, and Rouquier.
• To develop a general framework for understanding how categorical actions of Lie algebras behave under deformation of their spectra, particularly through the construction of 'unfurlings' of Dynkin diagrams.
• To extend the reach of categorification techniques beyond highest and lowest weight representations by constructing non-degenerate categorifications of tensor products of highest and lowest weight modules.

Proposed method

• Employing deformation theory of categorical representations, the authors consider a one-parameter family of representations where the eigenvalues of the dot generators (yk) are deformed from nilpotent to generic values.
• By analyzing the spectrum of the dot operators, they construct a new graph ˜I and associated Kac-Moody algebra ˜g, called an 'unfurling' of g, which captures the structure of the deformed representation.
• The key technical tool is upper semicontinuity of dimension: if the deformed representation has the expected dimension at the generic point, then the original representation cannot be smaller.
• They define a completion bU of the 2-category U via the GC topology, enabling control over morphism spaces and allowing the use of power series and formal deformations.
• The construction relies on factoring the polynomials Qij(x,y) defining the KLR algebra into products involving roots of unity, which allows the lifting of spectral data to the covering graph ˜I.
• They prove that the map σ: (i,u) ↦ (i, ζ^di u) is an admissible automorphism of ˜I, and that the quotient ˜I/σ recovers the original Cartan datum, establishing a precise link between the deformed and original structures.

Experimental results

Research questions

• RQ1 Does the 2-category U(g) categorifying a symmetrizable Kac-Moody algebra g have a basis isomorphic to that of ˙U(g), as conjectured by Khovanov, Lauda, and Rouquier?
• RQ2 Can non-degeneracy of U(g) be established for all Cartan data, including infinite-type and non-finite-type cases, beyond the previously known finite-type results?
• RQ3 How do categorical actions of Lie algebras behave under deformation of the spectrum of the dot generators, and what structures emerge in the deformed setting?
• RQ4 Is there a general mechanism—'unfurling'—that relates the representation theory of a Lie algebra g to that of a larger algebra ˜g via spectral lifting, and how does this relate to folding of Dynkin diagrams?
• RQ5 Can tensor product categorifications of highest and lowest weight modules be made non-degenerate for weights outside the open Tits cone, thereby enabling full non-degeneracy of U(g)?

Key findings

• The paper proves Theorem A: for any commutative ring k and any Cartan datum with homogeneous polynomials Qij, the 2-category U(g) is non-degenerate and its Grothendieck group is isomorphic to ˙U(g).
• The authors construct a new 2-category ˜U(˜g) via 'unfurling' the original KLR algebra by lifting the spectrum of dot generators to roots of unity, resulting in a larger Lie algebra ˜g with a natural categorical action.
• The deformation technique shows that if the deformed representation has the expected dimension at the generic point, then the original representation at the special point cannot be smaller, thus proving non-degeneracy.
• The construction of the graph ˜I as a covering of the original Dynkin diagram I via roots of unity provides a systematic way to lift spectral data and recover the original structure upon quotienting by an admissible automorphism σ.
• The paper establishes that the power series P◦,uu′ij(x,y) relating the deformed and original polynomials encode the geometric coefficients of the unfurled graph, ensuring compatibility of the categorical action.
• The method successfully extends non-degeneracy to all weights, including those outside the open Tits cone, by using tensor products of highest and lowest weight representations as test modules.