Branching laws for tensor modules over classical locally finite Lie algebras

TL;DR

This paper studies how tensor modules over classical infinite-dimensional Lie algebras decompose when restricted to certain subalgebras, providing explicit branching laws for a broad class of embeddings.

Contribution

It introduces the notion of general tensor type embeddings and derives detailed branching laws for tensor modules over classical locally finite Lie algebras.

Findings

Determines socle filtrations of tensor modules under embeddings.

Provides explicit branching laws for all embeddings except gl(infty).

Extends previous classification of embeddings by Dimitrov and Penkov.

Abstract

Let g' and g be isomorphic to any two of the Lie algebras gl(infty), sl(infty), sp(infty), and so(infty). Let M be a simple tensor g-module. We introduce the notion of an embedding of g' into g of general tensor type and derive branching laws for triples g', g, and M, where the embedding of g' into g is of general tensor type. More precisely, since M is in general not semisimple as a g'-module, we determine the socle filtration of M over g'. Due to the description of embeddings of classical locally finite Lie algebras given by Dimitrov and Penkov, our results hold for all possible embeddings of g' into g unless g' is isomorphic to gl(infty).