$\mathfrak{sl}_2$ 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

TL;DR

This paper extends the construction of $rak{sl}_2$ subalgebras in rank 2 symmetric hyperbolic Kac-Moody Lie algebras, focusing on nilpositive elements within the span of real root vectors, and explores their module actions.

Contribution

It introduces a new series of $rak{sl}_2$ subalgebras in hyperbolic Kac-Moody Lie algebras, extending previous finite-dimensional theories.

Findings

Constructed new $rak{sl}_2$ subalgebras in rank 2 hyperbolic Kac-Moody algebras.

Analyzed $rak{sl}_2$ modules arising from these subalgebras.

Extended finite-dimensional nilpotent orbit theory to hyperbolic Kac-Moody context.

Abstract

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 $\mathfrak{sl}_2$ 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.