Extended Affine Root System IV (Simply-Laced Elliptic Lie Algebras)

Abstract

Let $(R,G)$ be a pair consisting of an elliptic root system $R$ and a marking $G$ of $R$. Assume that the attached elliptic Dynkin diagram $\Gamma (R,G)$ is simply-laced. To the simply-laced elliptic root system, we associate three Lie algebras, explained in 1), 2) and 3) below. The main result of the present paper is to show that all three algebras are isomorphic.

1. The first one, studied in $\S$3, is the subalgebra $\mathfrak{g}(R)$ generated by the highest vector $e^{\alpha }$ for all $\alpha \in R$ in the quotient Lie algebra $V_{Q(R)}/D{V}_{Q(R)}$ of the lattice vertex algebra attached to the elliptic root lattice $Q(R)$.
2. The second algebra $\mathfrak{e}({\Gamma }_{\mathrm{ell}})$, studied in $\S$4, is presented by the Chevalley generators and the generalized Serre relations attached to the elliptic Dynkin diagram ${\Gamma }_{\mathrm{ell}}=\Gamma (R,G)$.
3. The third algebra ${\mathfrak{h}}_{\mathrm{af}}^{{\mathbb{Z}}^{\prime }}\ast {\mathfrak{g}}_{\mathrm{af}}$, studied in $\S$5, is defined as an amalgamation of an affine Heisenberg algebra and an affine Kac–Moody algebra together with the finite amalgamation relations.