[Paper Review] Simple bounded weight modules of $\displaystyle{\mathfrak{sl}(\infty)}$, $\displaystyle{\mathfrak{o}(\infty)}$, $\displaystyle{\mathfrak{sp}(\infty)}$

Dimitar Grantcharov, Ivan Penkov|arXiv (Cornell University)|Jul 5, 2018

Algebraic structures and combinatorial models10 references4 citations

TL;DR

This paper classifies simple bounded weight modules for the infinite-dimensional Lie algebras $σλ(\infty)$, $ϴ(\infty)$, and $σπ(\infty)$, proving that nonintegrable modules are multiplicity-free and computing their annihilators in the universal enveloping algebra. The key result is a complete classification of such modules and their primitive ideals, with annihilators explicitly determined via direct limit constructions and localization techniques.

ABSTRACT

We classify the simple bounded weight modules of ${\mathfrak{sl}(\infty})$, ${\mathfrak{o}(\infty)}$ and ${\mathfrak{sp}(\infty)}$, and compute their annihilators in $U({\mathfrak{sl}(\infty}))$, $U({\mathfrak{o}(\infty))}$, $U({\mathfrak{sp}(\infty))}$, respectively.

Motivation & Objective

• To classify all simple bounded weight modules for the direct limit Lie algebras $σλ(\infty)$, $ϴ(\infty)$, and $σπ(\infty)$.
• To compute the annihilators of these modules in the universal enveloping algebras $U(σλ(\infty))$, $U(ϴ(\infty))$, and $U(σπ(\infty))$.
• To extend the classification of bounded modules from finite-dimensional Lie algebras to the infinite-dimensional case, particularly for $σλ(\infty)$, $ϴ(\infty)$, and $σπ(\infty)$.
• To resolve a long-standing open problem in representation theory of finitary Lie algebras by providing a complete and explicit classification of simple bounded weight modules.

Proposed method

• Utilizes the theory of weight modules over finite-dimensional Lie algebras as a foundation, drawing on results from Fernando, Mathieu, and others.
• Applies localization techniques and twisted localization to analyze the structure of bounded weight modules over $σλ(\infty)$, $ϴ(\infty)$, and $σπ(\infty)$.
• Employs direct limit constructions to lift results from finite-dimensional $σλ(n+1)$, $ϴ(2n+1)$, $σπ(2n)$ to the infinite-dimensional case.
• Uses the Shale-Weil representation and its properties to identify key primitive ideals, particularly for $σπ(\infty)$.
• Applies the concept of multiplicity-freeness to classify nonintegrable modules, showing that all nonintegrable simple bounded modules are multiplicity-free.
• Computes annihilators via intersection with finite-dimensional universal enveloping algebras $U_n$, showing that $(\operatorname{Ann}M)\cap U_n$ stabilizes and defines the full annihilator in $U(\mathfrak{g})$.

Experimental results

Research questions

• RQ1 What is the complete classification of simple bounded weight modules for $σλ(\infty)$, $ϴ(\infty)$, and $σπ(\infty)$?
• RQ2 What are the annihilators of these modules in the universal enveloping algebra $U(\mathfrak{g})$?
• RQ3 Are nonintegrable simple bounded weight modules multiplicity-free, and how does this simplify the classification?
• RQ4 How do the annihilators of bounded modules behave under direct limit constructions from finite-dimensional algebras?
• RQ5 What is the structure of the primitive ideals arising from bounded weight modules in $U(\mathfrak{g})$?

Key findings

• All nonintegrable simple bounded weight modules of $σλ(\infty)$ and $σπ(\infty)$ are multiplicity-free, confirming a key insight from Dimitrov’s earlier work.
• The annihilator of the simple bounded nonintegrable $σπ(\infty)$-module $L(\frac{1}{2}, \frac{1}{2}, \dots)$ is the primitive ideal $I_{\rm sw} = \underrightarrow{\lim}\, J_n$, where $J_n$ is the annihilator of the Shale-Weil module in $U(\mathfrak{sp}(2n))$.
• For $σλ(\infty)$, the annihilator of $\Lambda_A^{\infty/2}V$ is isomorphic to $I(0,1,\emptyset,\emptyset)$, and the annihilator of $S_A^\infty V$ or $S_A^\infty V_*$ is isomorphic to $I(1,0,\emptyset,\emptyset)$.
• For $σλ(\infty)$, the annihilator of $S^\lambda V$ is isomorphic to $I(0,0,\lambda,\emptyset)$, and the annihilator of $S^\mu V_*$ is isomorphic to $I(0,0,\emptyset,\mu)$.
• For $ϴ(\infty)$, the annihilator of any nontrivial simple bounded integrable module $M \not\simeq V$ is nonzero and independent of $M$, and differs from the annihilator of the trivial module $V$.
• The annihilator of the standard module $V$ for $σπ(\infty)$ is not equal to $I_{\rm sw}$, showing that $I_{\rm sw}$ is not the annihilator of the trivial representation.