Irreducible module decompositions of rank 2 symmetric hyperbolic Kac-Moody Lie algebras by $\mathfrak{sl}_2$ subalgebras which are generalizations of principal $\mathfrak{sl}_2$ subalgebras

TL;DR

This paper constructs and classifies certain $rak{sl}_2$ subalgebras within rank 2 symmetric hyperbolic Kac-Moody Lie algebras, demonstrating their irreducible decomposition and analyzing the multiplicities of series components.

Contribution

It introduces generalized $rak{sl}_2$ subalgebras for rank 2 hyperbolic Kac-Moody algebras and classifies their irreducible decompositions.

Findings

Decomposition of rank 2 hyperbolic Kac-Moody algebras under these subalgebras.

Classification of irreducible components and their multiplicities.

Identification of unitary principal and complementary series.

Abstract

There exist principal $\mathfrak{sl}_2$ subalgebras for hyperbolic Kac-Moody Lie algebras. In the case of rank 2 symmetric hyperbolic Kac-Moody Lie algebras, certain $\mathfrak{sl}_2$ subalgebras are constructed. These subalgebras are generalizations of principal $\mathfrak{sl}_2$ subalgebras. We show that the rank 2 symmetric hyperbolic Kac-Moody Lie algebras themselves are irreducibly decomposed under the action of this $\mathfrak{sl}_2$ subalgebras. Furthermore, we classify irreducible components of the decomposition. In particular, we obtain multiplicities of unitary principal series and complementary series.