[Paper Review] sl(N)-Web categories

Marco Mackaay, Y. Yonezawa|arXiv (Cornell University)|Jun 26, 2013

Algebraic structures and combinatorial models28 references17 citations

TL;DR

This paper constructs a categorification of quantum skew Howe duality using colored $σρ_N$-matrix factorizations, establishing a 2-representation of categorified quantum $σρ_m$ on a web category. The key result is that the Karoubi envelope of this web category is equivalent to the category of finite-dimensional graded projective modules over a level-$N$ cyclotomic KLR algebra, categorically realizing the web space as a Grothendieck group.

ABSTRACT

In this paper we use colored sl(N)-matrix factorizations, due to Wu and Y.Y., in order to categorify part of the quantum skew Howe duality defined by Cautis, Kamnitzer and Morrison. In particular, we define web categories and 2-representations of Khovanov and Lauda's categorical quantum sl(m) on them. We show that each such web category is equivalent to the category of finite dimensional graded projective modules over a certain level N cyclotomic Khovanov-Lauda-Rouquier algebra.

Motivation & Objective

• To categorify the quantum skew Howe duality between $σρ_m$ and $σρ_N$ using matrix factorizations.
• To define a 2-representation of categorified quantum $σρ_m$ on a graded web category.
• To establish an equivalence between the Karoubi envelope of the web category and the category of finite-dimensional graded projective modules over a level-$N$ cyclotomic KLR algebra.
• To generalize Khovanov's arc algebras ($N=2$) and Pan-Tubbenhauer-Mackaay web algebras ($N=3$) to arbitrary $N$.

Proposed method

• Construct a 2-functor $Γ_{m,d,N}$ from Khovanov-Lauda's categorified quantum $σρ_m$ to a 2-category of colored $σρ_N$-matrix factorizations.
• Define a graded web category $ω_{Λ}^{∘}$ as a direct sum over web spaces indexed by weight $Λ = N\ell \omega_\ell$, with $d = N\ell$.
• Use matrix factorizations associated to $σρ_N$-webs to define 1-morphisms and 2-morphisms in the 2-category $ω_{m,d,N}$.
• Define the action of categorified $σρ_m$ on the web category via gluing of ladders and tensoring with matrix factorizations.
• Prove that the resulting 2-representation is strong and universal via Rouquier's universality proposition.
• Establish an isomorphism between the split Grothendieck group of the Karoubi envelope $ω_{Λ}^{∘}$ and the original web space $W_{Λ}$ via a non-degenerate $q$-sesquilinear form.

Experimental results

Research questions

• RQ1 Can the colored $σρ_N$-matrix factorizations be used to categorify the quantum skew Howe duality of Cautis-Kamnitzer-Morrison?
• RQ2 Does the 2-representation of categorified quantum $σρ_m$ on the web category $ω_{Λ}^{∘}$ extend to a strong 2-representation?
• RQ3 Is the Karoubi envelope $ω_{Λ}^{∘}$ equivalent to the category of finite-dimensional graded projective modules over the level-$N$ cyclotomic KLR algebra $R_{Λ}$?
• RQ4 How does the web category relate to the geometry of Springer varieties and arc algebras for $N=2$?
• RQ5 Can this framework be extended to define a complete set of relations for $σρ_N$-foams for $N \geq 4$?

Key findings

• The 2-functor $Γ_{m,d,N}$ provides a categorification of quantum skew Howe duality, mapping categorified $σρ_m$ to a 2-category of $σρ_N$-matrix factorizations.
• The web category $ω_{Λ}^{∘}$ carries a well-defined strong 2-representation of categorified quantum $σρ_m$ via matrix factorization gluing.
• The Karoubi envelope $ω_{Λ}^{∘}$ is equivalent to the category of finite-dimensional graded projective modules over the level-$N$ cyclotomic KLR algebra $R_{Λ}$.
• The split Grothendieck group $K_0^q(\dot{\mathcal{W}}_{Λ}^{∘})$ is isomorphic to the original web space $W_{Λ}$, with the isomorphism given by the $q$-dimension map.
• The web category decomposes into blocks, each equivalent to the category of finite-dimensional graded projective modules over an $σρ_N$-web algebra.
• For $N=2$, the web algebra is Khovanov's arc algebra; for $N=3$, it generalizes the construction of Pan-Tubbenhauer-Mackaay, and the framework extends to arbitrary $N$.