Tensor representations of classical locally finite Lie algebras

TL;DR

This paper investigates the structure of tensor representations of infinite-dimensional classical Lie algebras, revealing their non-semisimple nature and explicitly describing their composition factors and indecomposable components.

Contribution

It provides a detailed analysis of tensor representations of infinite-dimensional classical Lie algebras, including their Jordan-Holder constituents and socle filtrations, which was not previously known.

Findings

Tensor representations of infinite-dimensional Lie algebras are not semisimple.

Explicit descriptions of Jordan-Holder constituents and socle filtrations.

Identification of indecomposable direct summands.

Abstract

We study the structure of tensor representations of the classical infinite-dimensional locally finite Lie algebras $gl_\infty$, $sl_\infty$, $sp_\infty$ and $so_\infty$. In contrast with the finite-dimensional case, these tensor representations are not semisimple. We explicitly describe their Jordan-Holder constituents, socle filtrations, and indecomposable direct summands.