[Paper Review] Categorical representations, KLR algebras and Koszul duality

Ruslan Maksimau|arXiv (Cornell University)|Dec 15, 2015

Algebraic structures and combinatorial models3 references3 citations

TL;DR

This paper establishes that in the parabolic category $Σ\mathcal{O}$ for affine ${\mathfrak{gl}}_N$ at level $-N-e$, the categorical action of $\widetilde{\mathfrak{sl}}_e$ via functors $E$ and $F$ restricts to a subcategory $\mathbf{A}$ that categorifies higher-level Fock space. The key result is that $E$ and $F$ on $\mathbf{A}$ are Koszul dual to Zuckerman functors, proven via a decomposition of $F$ at level $-N-e-1$ and an isomorphism between KLR algebras associated to $A_{e-1}^{(1)}$ and a subquotient of $A_e^{(1)}$. The construction relies on categorical representation theory and KLR algebra techniques.

ABSTRACT

The parabolic category $\mathcal O$ for affine ${\mathfrak{gl}}_N$ at level $-N-e$ admits a structure of a categorical representation of $\widetilde{\mathfrak{sl}}_e$ with respect to some endofunctors $E$ and $F$. This category contains a smaller category $\mathbf A$ that categorifies the higher level Fock space. We prove that the functors $E$ and $F$ in the category $\mathbf A$ are Koszul dual to Zuckerman functors. The key point of the proof is to show that the functor $F$ for the category $\mathbf A$ at level $-N-e$ can be decomposed in terms of components of the functor $F$ for the category $\mathbf A$ at level $-N-e-1$. To prove this, we use the approach of categorical representations. We prove a general fact about categorical representations: a category with an action of $\widetilde{\mathfrak sl}_{e+1}$ contains a subcategory with an action of $\widetilde{\mathfrak sl}_{e}$. To prove this claim, we construct an isomorphism between the KLR algebra associated with the quiver $A_{e-1}^{(1)}$ and a subquotient of the KLR algebra associated with the quiver $A_{e}^{(1)}$.

Motivation & Objective

• To establish a categorical representation of $\widetilde{\mathfrak{sl}}_e$ on the parabolic category $\mathcal{O}$ for affine ${\mathfrak{gl}}_N$ at level $-N-e$.
• To identify a subcategory $\mathbf{A}$ within this category that categorifies higher-level Fock space.
• To prove that the functors $E$ and $F$ on $\mathbf{A}$ are Koszul dual to Zuckerman functors.
• To construct a general mechanism for restricting categorical $\widetilde{\mathfrak{sl}}_{e+1}$-actions to $\widetilde{\mathfrak{sl}}_e$-actions via KLR algebra subquotients.
• To demonstrate that the functor $F$ at level $-N-e$ decomposes in terms of components from level $-N-e-1$.

Proposed method

• Use categorical representation theory to define functors $E$ and $F$ acting on the parabolic category $\mathcal{O}$ at level $-N-e$.
• Identify the subcategory $\mathbf{A}$ as the categorification of higher-level Fock space via its $\widetilde{\mathfrak{sl}}_e$-action.
• Prove that the action of $F$ on $\mathbf{A}$ at level $-N-e$ decomposes into components arising from the $F$-functor at level $-N-e-1$, using level-shifting techniques.
• Construct an explicit isomorphism between the KLR algebra for the affine quiver $A_{e-1}^{(1)}$ and a subquotient of the KLR algebra for $A_e^{(1)}$, establishing the link between representation categories.
• Apply this isomorphism to show that the categorical $\widetilde{\mathfrak{sl}}_e$-action on $\mathbf{A}$ arises from a restriction of a larger $\widetilde{\mathfrak{sl}}_{e+1}$-action.
• Use Koszul duality to relate the functors $E$ and $F$ on $\mathbf{A}$ to Zuckerman functors, completing the duality statement.

Experimental results

Research questions

• RQ1 How does the categorical $\widetilde{\mathfrak{sl}}_e$-action on the parabolic category $\mathcal{O}$ for affine ${\mathfrak{gl}}_N$ at level $-N-e$ restrict to the subcategory $\mathbf{A}$ that categorifies higher-level Fock space?
• RQ2 What is the precise relationship between the functors $E$ and $F$ on $\mathbf{A}$ and Zuckerman functors in the context of Koszul duality?
• RQ3 Can the $F$-functor on $\mathbf{A}$ at level $-N-e$ be decomposed into components from the $F$-functor at level $-N-e-1$?
• RQ4 Does a categorical $\widetilde{\mathfrak{sl}}_{e+1}$-action on a category necessarily contain a subcategory with an induced $\widetilde{\mathfrak{sl}}_e$-action?
• RQ5 Is there a natural isomorphism between the KLR algebra of type $A_{e-1}^{(1)}$ and a subquotient of the KLR algebra of type $A_e^{(1)}$?

Key findings

• The subcategory $\mathbf{A}$ inside the parabolic category $\mathcal{O}$ at level $-N-e$ categorifies higher-level Fock space via its $\widetilde{\mathfrak{sl}}_e$-action.
• The functors $E$ and $F$ on $\mathbf{A}$ are Koszul dual to Zuckerman functors, establishing a deep duality in the representation-theoretic framework.
• The $F$-functor on $\mathbf{A}$ at level $-N-e$ decomposes into components derived from the $F$-functor at level $-N-e-1$, enabling inductive control over the action.
• A general construction shows that any category with a categorical $\widetilde{\mathfrak{sl}}_{e+1}$-action contains a subcategory with an induced $\widetilde{\mathfrak{sl}}_e$-action.
• An explicit isomorphism is constructed between the KLR algebra of type $A_{e-1}^{(1)}$ and a subquotient of the KLR algebra of type $A_e^{(1)}$, providing the algebraic backbone for the duality.