[Paper Review] On Categories $\mathcal{O}$ for Root-Reductive Lie Algebras

Thanasin Nampaisarn|arXiv (Cornell University)|Nov 30, 2017

Advanced Algebra and Geometry10 references3 citations

TL;DR

This paper introduces an analogue of the Bernstein-Gel'fand-Gel'fand category $\mathcal{O}$ for infinite-dimensional root-reductive Lie algebras over an algebraically closed field of characteristic zero. By focusing on Dynkin Borel subalgebras, it establishes a well-behaved category $\bar{\mathcal{O}}$ with Verma modules, composition series analogues, and a generalized Kazhdan-Lusztig theory, while proving a version of BGG reciprocity for truncated subcategories despite the absence of injective objects.

ABSTRACT

Let $\mathfrak{g}$ be a root-reductive Lie algebra over an algebraically closed field $\mathbb{K}$ of characteristic $0$ with a splitting Borel subalgebra $\mathfrak{b}$ containing a splitting maximal toral subalgebra $\mathfrak{h}$. We study the category $\bar{\mathcal{O}}$ consisting of all $\mathfrak{h}$-weight $\mathfrak{g}$-modules which are locally $\mathfrak{b}$-finite and have finite-dimensional $\mathfrak{h}$-weight spaces. The focus is on very special Borel subalgebras called the Dynkin Borel subalgebras. This paper serves as an initial passage to the understanding of categories $\mathcal{O}$ for infinite-dimensional root-reductive Lie algebras.

Motivation & Objective

• To extend the classical BGG category $\mathcal{O}$ to infinite-dimensional root-reductive Lie algebras, which are direct limits of reductive Lie algebras with specific restrictions.
• To define and study a generalized category $\bar{\mathcal{O}}$ consisting of $\mathfrak{h}$-semisimple modules with locally finite $\mathfrak{b}$-action and finite-dimensional $\mathfrak{h}$-weight spaces.
• To focus on Dynkin Borel subalgebras—special Borel subalgebras that endow $\bar{\mathcal{O}}$ with enhanced structural properties such as the existence of Verma modules and composition series analogues.
• To generalize Kazhdan-Lusztig theory to $\bar{\mathcal{O}}$ and establish a version of BGG reciprocity for truncated subcategories despite the lack of injective objects.

Proposed method

• Define the extended category $\bar{\mathcal{O}}$ for a root-reductive Lie algebra $\mathfrak{g}$ with respect to a splitting Borel subalgebra $\mathfrak{b}$ containing a splitting maximal toral subalgebra $\mathfrak{h}$, requiring $\mathfrak{h}$-semisimplicity, locally finite $\mathfrak{b}$-action, and finite-dimensional $\mathfrak{h}$-weight spaces.
• Focus on Dynkin Borel subalgebras, which are characterized by having a root system that is a union of finite root systems with a specific ordering, enabling stronger structural results.
• Utilize the structure of the derived ideal $[\mathfrak{g}, \mathfrak{g}]$, which is a direct sum of finite-dimensional simple Lie algebras and simple finitary Lie algebras like $\mathfrak{sl}_\infty$, $\mathfrak{so}_\infty$, $\mathfrak{sp}_\infty$.
• Construct Verma modules as induced modules from $\mathfrak{b}$-modules, and show they are objects in $\bar{\mathcal{O}}$ when $\mathfrak{b}$ is a Dynkin Borel subalgebra.
• Analyze the block decomposition of $\bar{\mathcal{O}}$ via the action of the Weyl group and the equivalence relation on weights induced by the Weyl groupoid.
• Apply techniques from [RCW] to establish BGG reciprocity for truncated subcategories of $\bar{\mathcal{O}}$, circumventing the absence of injective objects in the full category.

Experimental results

Research questions

• RQ1 How can the classical BGG category $\mathcal{O}$ be generalized to infinite-dimensional root-reductive Lie algebras?
• RQ2 What structural properties does the category $\bar{\mathcal{O}}$ possess when defined with respect to a Dynkin Borel subalgebra?
• RQ3 Can Kazhdan-Lusztig theory be extended to $\bar{\mathcal{O}}$ for root-reductive Lie algebras?
• RQ4 How does the block structure of $\bar{\mathcal{O}}$ reflect the root system and Weyl groupoid action?
• RQ5 Is there a version of BGG reciprocity in $\bar{\mathcal{O}}$ despite the lack of injective objects?

Key findings

• The category $\bar{\mathcal{O}}$ with respect to a Dynkin Borel subalgebra contains all Verma modules, which are fundamental in the representation theory of Lie algebras.
• Every object in $\bar{\mathcal{O}}$ admits an analogue of a composition series, with finite-length quotients that are irreducible or Verma modules.
• The Kazhdan-Lusztig theory generalizes to $\bar{\mathcal{O}}$, allowing the use of Kazhdan-Lusztig polynomials to describe composition factors in Verma modules.
• The block decomposition of $\bar{\mathcal{O}}$ is controlled by the Weyl groupoid action on the weight lattice, with blocks indexed by orbits under this action.
• Although $\bar{\mathcal{O}}$ lacks injective objects, a version of BGG reciprocity holds for its truncated subcategories, as established via the method of [RCW].
• The derived ideal $[\mathfrak{g}, \mathfrak{g}]$ of a root-reductive Lie algebra $\mathfrak{g}$ is a direct sum of finite-dimensional simple Lie algebras and simple finitary Lie algebras, each occurring with at most countable multiplicity.