Regular subalgebras of affine Kac-Moody algebras

Closed sets of real roots in Kac-Moody root systems

In this note, we provide a complete description of the closed sets of real roots in a Kac-Moody root system.

Lattice of dominant weights of affine Kac-Moody algebras

The dual space of the Cartan subalgebra in a Kac-Moody algebra has a partial ordering defined by the rule that two elements are related if and only if their difference is a non-negative or non-positive integer linear combination of simple roots. In this paper, we study the subposet formed by dominant weights in affine Kac-Moody algebras. We give a more explicit description of the covering relations in this poset. We also study the structure of basic cells in this poset of dominant weights for untwisted affine Kac-Moody algebras of type A.

Real forms and finite order automorphisms of affine Kac-Moody algebras - an outline of a new approach

We outline a new approach to classify real forms and automorphisms of finite order of affine Kac-Moody algebras.

π-系统与秩2 Kac-Moody李代数的嵌入问题

$π$-systems and the embedding problem for rank $2$ Kac-Moody Lie algebras

本文研究了秩2 Kac-Moody李代数的π-系统分类及其在嵌入问题中的应用，证明了大部分π-系统线性无关，并提供了相关子代数的构造方法。

Toward a Jacobson--Morozov theorem for Kac--Moody Lie algebras

For a finite-dimensional semisimple Lie algebra $\mathfrak{g}$, the Jacobson--Morozov theorem gives a construction of subalgebras $\mathfrak{sl}_2 \subset \mathfrak{g}$ corresponding to nilpotent elements of $\mathfrak{g}$. In this note, we propose an extension of the Jacobson--Morozov theorem to the symmetrizable Kac--Moody setting and give a proof of this generalization in the case of rank two hyperbolic Kac--Moody algebras. We also give a proof for an arbitrary symmetrizable Kac--Moody algebra under some stronger restrictions.

A combinatorial approach to root multiplicities of rank 2 hyperbolic Kac-Moody algebras

In this paper we study root multiplicities of rank 2 hyperbolic Kac-Moody algebras using the combinatorics of Dyck paths.

Automorphisms and derivations of Borel subalgebras and their nilradicals in Kac-Moody algebras

In this paper, we determine derivations of Borel subalgebras and their derived subalgebras called nilradicals, in Kac-Moody algebras (and contragredient Lie algebras) over any field of characteristic 0; and we also determine automorphisms of those subalgebras in symmetrizable Kac-Moody algebras. The results solve a conjecture posed by R. V. Moody about 30 years ago which generalizes a result by B. Kostant and which is discussed by A. Fialowski using Lie algebra cohomology in case of affine type.

Laplacian spectrum for the nilpotent Kac-Moody Lie algebras

We prove that the maximal nilpotent subalgebra of a Kac-Moody Lie algebra has an (essentially unique) Euclidean metric with respect to which the Laplace operator in the chain complex is scalar on each component of a given degree. Moreover, both the Lie algebra structure and the metric are uniquely determined by this property.

A certain class of standard subalgebras of affine Kac-Moody algebras

The aim of this paper is to extend the theory of standard subalgebras of finite dimensional simple Lie algebras to infinite dimensional Lie algebras. We construct and characterize a class of standard subalgebras of affine Kac-Moody algebra.

An algorithm for computing root multiplicities in Kac-Moody algebras

Root multiplicities encode information about the structure of Kac-Moody algebras, and appear in applications as far-reaching as string theory and the theory of modular functions. We provide an algorithm based on the Peterson recurrence formula to compute multiplicities, and argue that it is more efficient than the naive algorithm.

Some branching formulas for Kac--Moody Lie algebras

In this paper we give some branching rules for the fundamental representations of Kac--Moody Lie algebras associated to $T$-shaped graphs. These formulas are useful to describe generators of the generic rings for free resolutions of length three described in [JWm16]. We also make some conjectures about the generic rings.

Absolutely indecomposable representations and Kac-Moody Lie algebras (with an appendix by Hiraku Nakajima)

A conjecture of Kac states that the polynomial counting the number of absolutely indecomposable representations of a quiver over a finite field with given dimension vector has positive coefficients and furthermore that its constant term is equal to the multiplicity of the corresponding root in the associated Kac-Moody Lie algebra. In this paper we prove these conjectures for indivisible dimension vectors.