Related papers: Maximal rank root subsystems of hyperbolic root sy…

We completely classify the real root subsystems of root systems of loop algebras of Kac-Moody Lie algebras. This classification involves new notions of "admissible subgroups" of the coweight lattice of a root system $\Psi$, and "scaling…

For the affine and hyperbolic root system the symmetric part of the centralizers of root subgroups in the corresponding Steinberg groups are calculated. In the affine case the corresponding root subsystems can be computed in term of the…

We show that the rank 10 hyperbolic Kac-Moody algebra $E_{10}$ contains every simply laced hyperbolic Kac-Moody algebra as a Lie subalgebra. Our method is based on an extension of earlier work of Feingold and Nicolai.

That finite-dimensional simple Lie algebras over the complex numbers can be classified by means of purely combinatorial and geometric objects such as Coxeter-Dynkin diagrams and indecomposable irreducible root systems, is arguably one of…

In this paper we study rank two symmetric hyperbolic Kac-Moody algebras H(a) and their automorphic correction in terms of Hilbert modular forms. We associate a family of H(a)'s to the quadratic field Q(p) for each odd prime p and show that…

Using the coset construction, we compute the root multiplicities at level three for some hyperbolic Kac-Moody algebras including the basic hyperbolic extension of $A_1^{(1)}$ and $E_{10}$.

The group of automorphisms of the Kac Jordan superalgebra is described, and used to classify the maximal subalgebras.

We study the Weyl groups of hyperbolic Kac-Moody algebras of `over-extended' type and ranks 3, 4, 6 and 10, which are intimately linked with the four normed division algebras K=R,C,H,O, respectively. A crucial role is played by integral…

We use the theory of Clifford algebras and Vahlen groups to study Weyl groups of hyperbolic Kac-Moody algebras T_n^{++}, obtained by a process of double extension from a Cartan matrix of finite type T_n, whose corresponding generalized…

In this paper, we study a special class of indefinite Kac-Moody algebras. Based on the study of hyperbolic Kac-Moody algebras, we give the definition of $N_k$ type Kac-Moody algebras and study some properties of this special type Kac-Moody…

We look to gradations of Kac-Moody Lie algebras by Kac-Moody root systems with finite dimensional weight spaces. We extend, to general Kac-Moody Lie algebras, the notion of C-admissible pair as introduced by H. Rubenthaler and J. Nervi for…

Starting from Borcherds' fake monster Lie algebra we construct a sequence of six generalized Kac-Moody algebras whose denominator formulas, root systems and all root multiplicities can be described explicitly. The root systems decompose…

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…

We describe a new large class of Lorentzian Kac--Moody algebras. For all ranks, we classify 2-reflective hyperbolic lattices S with the group of 2-reflections of finite volume and with a lattice Weyl vector. They define the corresponding…

We give a criterion for a Dynkin diagram, equivalently a generalized Cartan matrix, to be symmetrizable. This criterion is easily checked on the Dynkin diagram. We obtain a simple proof that the maximal rank of a Dynkin diagram of compact…

This paper constructs the Vogan diagrams for hyperbolic Kac- Moody algebras, which have potential physical applications in cosmological billiards.

In analogy to the theory of nilpotent orbit in finite-dimensional semisimple Lie algebras, it is known that the principal $\mathfrak{sl}_2$ subalgebras can be constructed in hyperbolic Kac-Moody Lie algebras. We obtained a series of…

We discuss a general theory of Lorentzian Kac--Moody algebras which should be a hyperbolic analogy of the classical theories of finite-dimensional semi-simple and affine Kac-Moody algebras. First examples of Lorentzian Kac-Moody algebras…

Given a symmetrizable Kac-Moody algebra $\mathfrack{g}$, we study its $\pi$-systems, which are subsets of real roots, the pairwise differences of whose elements are not roots. Such systems arise as simple systems of regular subalgebras of…

Let $k$ be an arbitrary field. We classify the maximal reductive subgroups of maximal rank in any classical simple algebraic $k$-group in terms of combinatorial data associated to their indices. This result complements [S, 2022], which does…