Unified characterizations of minuscule Kac--Moody representations built from colored posets

Abstract

R.M. Green described structural properties that ``doubly infinite'' colored posets should possess so that they can be used to construct representations of most affine Kac--Moody algebras. These representations are analogs of the minuscule representations of the semisimple Lie algebras, and his posets (``full heaps'') are analogs of the finite minuscule posets. Here only simply laced Kac--Moody algebras are considered. Working with their derived subalgebras, we provide a converse to Green's theorem. Smaller collections of colored structural properties are also shown to be necessary and sufficient for such poset-built representations to be produced for smaller subalgebras, especially the ``Borel derived'' subalgebra. These developments lead to the formulation of unified definitions of finite and infinite colored minuscule and $d$-complete posets. This paper launches a program that seeks to extend the notion of ``minuscule representation'' to Kac--Moody algebras, and to classify such representations.