[Paper Review] Annihilators of simple tensor modules

Alexandru-Gabriel Sava|arXiv (Cornell University)|Jan 18, 2012

Algebraic structures and combinatorial models19 references3 citations

TL;DR

This paper investigates primitive ideals in the universal enveloping algebra of finitary locally finite infinite-dimensional Lie algebras, such as $\mathfrak{sl}(\infty,\mathbb{C})$, $\mathfrak{o}(\infty,\mathbb{C})$, and $\mathfrak{sp}(\infty,\mathbb{C})$. It proves that two simple tensor modules are isomorphic if and only if they have the same annihilator, establishing a complete classification of simple modules via their annihilators in this infinite-dimensional setting.

ABSTRACT

We study primitive ideals in the enveloping algebra of finitary locally finite infinite-dimensional complex Lie algebras. In particular we investigate the annihilators of the simple objects in the category of tensor modules. This category has been studied in \cite{PStyr} and \cite{PS}. Moreover we prove that a simple tensor module is completely determined by its annihilator and we introduce a partial order on pairs of Young diagrams that we use to describe the inclusions between the annihilators of the simple tensor modules.

Motivation & Objective

• To classify simple modules for finitary locally finite infinite-dimensional Lie algebras such as $\mathfrak{sl}(\infty,\mathbb{C})$, $\mathfrak{o}(\infty,\mathbb{C})$, and $\mathfrak{sp}(\infty,\mathbb{C})$ via their annihilators in the universal enveloping algebra.
• To address the challenge of classifying simple modules in infinite-dimensional Lie algebras where the center of the universal enveloping algebra is trivial, making classical Duflo theory inapplicable.
• To determine whether non-trivial primitive ideals exist in $\mathcal{U}(\mathfrak{g})$ for these algebras despite the center being trivial.
• To establish a bijection between isomorphism classes of simple tensor modules and their annihilators, resolving the fiber problem of the annihilator map.

Proposed method

• Utilizes the structure of tensor modules over locally finite Lie algebras, which are inductive limits of finite-dimensional representations of nested finite-dimensional subalgebras.
• Applies the theory of Gelfand-Tsetlin patterns and highest weight modules to analyze restrictions of modules to increasing subalgebras $\mathfrak{g}_j$.
• Employs an inductive procedure on highest weight vectors, tracking strict dominance of weight partitions $ (\lambda^k, \mu^k) \prec (\lambda^{k+1}, \mu^{k+1}) $, to construct a chain of weights.
• Uses the fact that the category of tensor modules is closed under restriction and that simple modules are determined by their highest weights in the limit.
• Applies results from integrable modules and locally simple Lie algebras to ensure the existence of highest weight vectors in the inductive limit.
• Leverages the uniqueness of the limit module $ V_{\lambda^s\mu^s} $ generated by a stable highest weight vector to conclude isomorphism to the original simple module.

Experimental results

Research questions

• RQ1 Do non-trivial primitive ideals exist in $\mathcal{U}(\mathfrak{g})$ for $\mathfrak{g} = \mathfrak{sl}(\infty,\mathbb{C})$, $\mathfrak{o}(\infty,\mathbb{C})$, or $\mathfrak{sp}(\infty,\mathbb{C})$, given that the center of $\mathcal{U}(\mathfrak{g})$ is trivial?
• RQ2 Can the isomorphism class of a simple tensor module be uniquely determined by its annihilator in $\mathcal{U}(\mathfrak{g})$?
• RQ3 Is there a finite chain of weight partitions $ (\lambda^k, \mu^k) $ such that the module stabilizes to a highest weight module in the inductive limit?
• RQ4 Does the restriction of a simple module to increasing subalgebras $\mathfrak{g}_j$ eventually stabilize to a module isomorphic to a highest weight module $V_{\lambda^s\mu^s}$?
• RQ5 Can the annihilator of a simple tensor module be used to reconstruct the module up to isomorphism?

Key findings

• The annihilator of a simple tensor module uniquely determines its isomorphism class: two simple tensor modules are isomorphic if and only if they have the same annihilator in $\mathcal{U}(\mathfrak{g})$.
• Despite the center of $\mathcal{U}(\mathfrak{g})$ being trivial for $\mathfrak{g} = \mathfrak{sl}(\infty,\mathbb{C})$, $\mathfrak{o}(\infty,\mathbb{C})$, or $\mathfrak{sp}(\infty,\mathbb{C})$, non-trivial primitive ideals exist.
• The inductive limit of highest weight modules $F_{n+p+q}^{(\lambda^k,\mu^k)_{n}}$ stabilizes to a simple module $V_{\lambda^s\mu^s}$, which is isomorphic to the original module $M$.
• A chain of strictly increasing weight partitions $ (\lambda^1, \mu^1) \prec (\lambda^2, \mu^2) \prec \cdots \prec (\lambda^s, \mu^s) \preceq (\lambda, \mu) $ exists, with length bounded by $ (\lambda_1 + \mu_1)^{p+q} $, ensuring termination.
• The procedure of lifting highest weight vectors through successive restrictions eventually stabilizes in case 1, meaning the highest weight vector generates a module isomorphic to $V_{\lambda^s\mu^s}$.
• The conclusion follows from the simplicity of $M$ and the fact that $M$ is generated by a single highest weight vector in the limit, hence $M \simeq V_{\lambda^s\mu^s}$.