Large annihilator category O for sl_{\infty}, o_{\infty}, sp_{\infty}

TL;DR

This paper develops a new category of modules for infinite-dimensional Lie algebras, characterized by a large annihilator condition, and analyzes its structure, including multiplicities of simple and standard objects.

Contribution

It introduces a universal highest weight category for infinite-dimensional Lie algebras with large annihilator conditions, independent of Borel subalgebra choices.

Findings

Constructed the large annihilator category  for ( ) and related algebras.

Computed multiplicities of simple objects in standard modules.

Analyzed the structure and injective objects within the category.

Abstract

We construct a new analogue of the BGG category $\mathcal O$ for the infinite-dimensional Lie algebras $\fg=\mathfrak{sl}(\infty),\mathfrak{o}(\infty), \mathfrak{sp}(\infty)$. A main difference with the categories studied in \cite{Nam} and \cite{CP} is that all objects of our category satisfy the large annihilator condition introduced in \cite{DPS}. Despite the fact that the splitting Borel subalgebras $\fb$ of $\fg$ are not conjugate, one can eliminate the dependency on the choice of $\fb$ and introduce a universal highest weight category $\mathcal {OLA}$ of $\fg$-modules, the letters $\mathcal{LA}$ coming from "large annihilator". The subcategory of integrable objects in $\mathcal {OLA}$ is precisely the category $\mathbb T_{\fg}$ studied in \cite{DPS}. We investigate the structure of $\mathcal {OLA}$, and in particular compute the multiplicities of simple objects in standard objects and the multiplicities of standard objects in indecomposable injectives.