Symmetric real-closed subsets of affine Kac-Moody Lie Algebras

Abstract

In this thesis, we study some important classification problems related to affine Kac--Moody Lie algebras. First, we address the combinatorial problem of classifying symmetric real-closed subsets of affine root systems (which are roots of affine Kac--Moody Lie algebras). Secondly, we understand the correspondence between these symmetric real-closed subsets and the regular subalgebras generated by them, using the aforementioned classification. Motivation for this work comes from the celebrated work of Dynkin (1952), where he classified the semi-simple subalgebras of a given finite-dimensional semisimple Lie algebra 𝔤. He introduced the notion of regular subalgebras in order to achieve this classification. It is not hard to see that the regular subalgebras of 𝔤 correspond to symmetric closed subsets of roots of 𝔤, so the problem of classifying regular subalgebras comes down to the combinatorial problem of classifying these subsets. The analogous problem of studying regular subalgebras of affine Kac--Moody Lie algebras was initiated by Anna Felikson et al. in 2008 and continued by Roy--Venkatesh in 2019, where they addressed the part of the combinatorial problem, namely provided the classification of maximal real closed subroot systems of affine root systems.

In the finite case, it is well known that symmetric closed subsets are in fact closed subroot systems. It is not true in general for affine root systems. So, it is natural to ask the following questions:

When a given symmetric real-closed subset of affine root system is a closed subroot system?

Is it possible to classify all symmetric real-closed subsets of affine root systems?

We give affirmative answers to these questions in this thesis. In the untwisted setting, we prove that any symmetric real-closed subset is indeed a closed subroot system. Twisted types need more careful analysis, since the finite part of a symmetric real-closed subset has two possibilities in these types, namely closed and semi-closed. For semi-closed case, the behavior of symmetric real-closed subsets varies for each type. We prove that there are three types of irreducible symmetric real-closed subsets for reduced real affine root systems, one of which did not appear in Roy--Venkatesh's work. We conclude our classification for the twisted case by determining explicitly when a symmetric closed subset is a closed subroot system, including the case when the ambient Lie algebra is the non-reduced affine Lie algebra A_2n^(2).

In the second part of the thesis, we explore the correspondence between symmetric real-closed subsets and regular subalgebras generated by them. Roy--Venkatesh proved that the map between closed subroot systems and the regular subalgebras generated by them is injective. We observe that it is not true in general when we extend this map to symmetric real-closed subsets. Let ψ be a real-closed subroot system. We describe the types of symmetric closed subsets that can appear in the fiber of the subalgebra generated by ψ. Moreover, we determine when these fibers are finite. In certain cases, we are able to describe very explicitly the defining parameters of the symmetric closed subsets appearing in the fiber.