A Survey on Invariant Cones in Infinite Dimensional Lie Algebras

TL;DR

This survey explores the duality between invariant convex sets and functions in infinite-dimensional Lie algebras, highlighting root decompositions and applications to various algebra classes.

Contribution

It systematically analyzes the duality and convexity properties in infinite-dimensional Lie algebras, introducing root decompositions as a key tool.

Findings

Duality between convex invariant sets and functions is established.

Root decompositions aid in understanding infinite-dimensional Lie algebra structures.

Open problems are formulated for future research.

Abstract

For the Lie algebra $\g$ of a connected infinite-dimensional Lie group~$G$, there is a natural duality between so-called semi-equicontinuous weak-*-closed convex Ad^*(G)-invariant subsets of the dual space $\g'$ and Ad(G)-invariant lower semicontinuous positively homogeneous convex functions on open convex cones in $\g$. In this survey, we discuss various aspects of this duality and some of its applications to a more systematic understanding of open invariant cones and convexity properties of coadjoint orbits. In particular, we show that root decompositions with respect to elliptic Cartan subalgebras provide powerful tools for important classes of infinite Lie algebras, such as completions of locally finite Lie algebras, Kac--Moody algebras and twisted loop algebras with infinite-dimensional range spaces. We also formulate various open problems.