${\mathfrak{sl}}_2$ Triples Whose Nilpositive Elements Are in a Space Which Is Spanned by the Real Root Vectors in Rank 2 Symmetric Hyperbolic Kac–Moody Lie Algebras

Abstract

In analogy with the theory of nilpotent orbits 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 ${\mathfrak{sl}}_2$ subalgebras in rank 2 symmetric hyperbolic Kac–Moody Lie algebras by extending the aforementioned construction. We present this result and also discuss ${\mathfrak{sl}}_2$ modules obtained by the action of the ${\mathfrak{sl}}_2$ subalgebras on the original Lie algebras.