Unified Characterizations of Minuscule Kac–Moody Representations Built from Colored Posets

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.

THE WADGE ORDER ON THE SCOTT DOMAIN IS NOT A WELL-QUASI-ORDER

We prove that the Wadge order on the Borel subsets of the Scott domain is not a well-quasi-order, and that this feature even occurs among the sets of Borel rank at most 2. For this purpose, a specific class of countable 2-colored posets $\mathbb{P}_{emb} $ equipped with the order induced by homomorphisms is embedded into the Wadge order on the $\Delta _2^0 $-degrees of the Scott domain. We then show that $\mathbb{P}_{emb} $ admits both infinite strictly decreasing chains and infinite antichains with respect to this notion of comparison, which therefore transfers to the Wadge order on the $\Delta _2^0 $-degrees of the Scott domain.