Decomposition of the rank 3 Kac-Moody Lie algebra $F$ with respect to the rank 2 hyperbolic subalgebra $Fib$

Abstract

In 1983 Feingold-Frenkel studied the structure of a rank 3 hyperbolic Kac-Moody algebra $\mathcal{F}$ containing the affine KM algebra $A^{(1)}_1$. In 2004 Feingold-Nicolai showed that $\mathcal{F}$ contains all rank 2 hyperbolic KM algebras with symmetric Cartan matrices, $A=\left[\begin{array}{cc} 2 & -a \\ -a & 2 \\ \end{array}\right], a\geq 3$. The case when $a=3$ is called $\mathcal{F}ib$ because of its connection with the Fibonacci numbers (Feingold 1980). Some important structural results about $\mathcal{F}$ come from the decomposition with respect to its affine subalgebra $A^{(1)}_1$. Here we study the decomposition of $\mathcal{F}$ with respect to its subalgebra $\mathcal{F}ib$. We find that $\mathcal{F}$ has a grading by $\mathcal{F}ib$-level, and prove that each graded piece, $\mathcal{F}ib(m)$ for $m\in\mathbb{Z}$, is an integrable $\mathcal{F}ib$-module. We show that for $|m|>2$, $\mathcal{F}ib(m)$ completely reduces as a direct sum of highest- and lowest-weight modules, and for $|m|\leq 2$, $\mathcal{F}ib(m)$ contains one irreducible non-standard quotient module $V^{\Lambda_m}=V(m)\Big/U(m)$. We then prove that the quotient $\mathcal{F}ib(m)\Big/V(m)$ completely reduces as a direct sum of one trivial module (on level 0), and standard modules. We give an algorithm for determining the inner multiplicities of any irreducible $\mathcal{F}ib$-module, in particular the non-standard modules on levels $|m|\leq 2$. We show that multiplicities of non-standard modules on levels $|m|=1,2$ do not follow the Kac-Peterson recursion, but instead appear to follow a recursion similar to Racah-Speiser. We then use results of Borcherds and Frenkel-Lepowsky-Meurman and construct vertex algebras from the root lattices of $\mathcal{F}$ and $\mathcal{F}ib$, and study the decomposition within this setting.