Inner Ideals of Simple Locally Finite Lie Algebras

TL;DR

This paper characterizes inner ideals in simple locally finite Lie algebras over algebraically closed fields of characteristic zero, linking their existence to diagonal type and describing regular inner ideals explicitly.

Contribution

It provides a complete description of inner ideals in simple locally finite Lie algebras, especially characterizing those of diagonal type and finitary simple Lie algebras.

Findings

Non-zero proper inner ideals exist iff the algebra is of diagonal type.

Regular inner ideals are characterized via ideals of the enveloping algebra.

Descriptions of regular inner ideals in finitary simple Lie algebras are provided.

Abstract

Inner ideals of simple locally finite dimensional Lie algebras over an algebraically closed field of characteristic 0 are described. In particular, it is shown that a simple locally finite dimensional Lie algebra has a non-zero proper inner ideal if and only if it is of diagonal type. Regular inner ideals of diagonal type Lie algebras are characterized in terms of left and right ideals of the enveloping algebra. Regular inner ideals of finitary simple Lie algebras are described.