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 -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.
Keywords
Cite
@article{arxiv.1808.05200,
title = {Unified characterizations of minuscule Kac--Moody representations built from colored posets},
author = {Michael C. Strayer},
journal= {arXiv preprint arXiv:1808.05200},
year = {2020}
}
Comments
28 pages, 3 figures, 2 tables. Updates to this version: Sections 13 and 14 are replaced by synopses in Section 12. Section 6 is reorganized. Each Fact is either deleted or relabeled as a Lemma and supplied with a proof. Many definitions and remarks are now displayed with numbered labels. More references to the examples are inserted. Numerous minor word changes are made