On some structural problems in the theory of infinite-dimensional Lie (super)algebras

Abstract

In this thesis, we examine two facets of infinite-dimensional Lie algebras. The first part focuses on the classification of $\pi$-systems in rank 2 Kac-Moody Lie algebras. In one of his seminal works, Dynkin classified the semisimple subalgebras of finite-dimensional semisimple Lie algebras, where the notion of $\pi$-systems served as a central tool. This concept continues to play a significant role in the study of infinite-dimensional Kac-Moody algebras, particularly in understanding the types of algebras that can be embedded within a given Kac-Moody algebra.

The limited understanding of $\pi$-systems and regular subalgebras beyond the affine case serves as the motivation for our study of $\pi$-systems in rank 2 Kac-Moody Lie algebras. We present an explicit classification of $\pi$-systems associated with these algebras and prove that, in most cases, they are linearly independent. This classification further allows us to identify the corresponding root generated subalgebras, thereby determining all possible Kac-Moody algebras that can be embedded in a rank 2 Kac-Moody algebra as subalgebras generated by real root vectors. Moreover, through explicit examples, we demonstrate that the classification of $\pi$-systems becomes significantly more challenging even in the rank 2 case when imaginary roots are permitted.

In the second part of this thesis, we investigate the root multiplicities of Borcherds-Kac-Moody (BKM) Lie superalgebras through their denominator identities, deriving explicit combinatorial formulas in terms of graph invariants associated with marked (quasi) Dynkin diagrams. A BKM Lie superalgebra may be regarded both as a natural generalization of a Kac-Moody Lie superalgebra and as a $\mathbb{Z}_2$-graded analogue of a Borcherds-Kac-Moody Lie algebra. The central motivation of this study is to analyze the root multiplicities encoded in the denominator identities of BKM Lie superalgebras and to obtain explicit formulas for these multiplicities via graph invariants of the underlying marked (quasi) Dynkin diagrams. This work extends earlier results of Venkatesh et al., Arunkumar et al., and Sushma et al., which uncovered connections between the chromatic polynomial of a graph and the root multiplicities of the associated Kac-Moody and BKM Lie algebras. A central notion in our approach is that of marked multi-colorings and their associated polynomials, which generalize chromatic polynomials and provide an effective framework for computing root multiplicities.

To pursue this program, we introduce partially commutative Lie superalgebras (PCLSAs) as a tool for studying certain roots of BKM Lie superalgebras. We give a direct combinatorial proof of their denominator identity, employing ideas from Viennot’s heap theory. Furthermore, we characterize the roots of PCLSAs and establish connections between their universal enveloping algebras and right-angled Coxeter groups, which in turn allows us to compute the Hilbert series of these groups explicitly.