GIM and Elliptic Lie algebras via Ringel--Hall Lie algebras

Abstract

For any symmetrizable generalized intersection matrix (GIM) $C$, we construct an acyclic valued quiver $(Q,\mathbf{d})$ endowed with an involution $\theta$. Let $\mathcal{D}$ be the bounded derived category of finite-dimensional representations of $(Q,\mathbf{d})$, and let $\Sigma$ stand for the suspension functor of $\mathcal{D}$. We show that the orbit category $\mathcal{D}/(\theta\circ\Sigma)$ carries a canonical triangulated structure and is $2$-periodic. Applying Peng--Xiao's construction to this orbit category, we prove that the GIM algebra $\operatorname{gim}(C)$ is isomorphic to the integral Ringel--Hall Lie algebra associated with $\mathcal{D}/(\theta\circ\Sigma)$. As a further application of the above machinery, we investigate elliptic Lie algebras of types $D_4^{(1,1)}$, $E_6^{(1,1)}$, $E_7^{(1,1)}$ and $E_8^{(1,1)}$. For each elliptic Dynkin diagram, we define a finite-dimensional algebra $A$ by taking an appropriate quotient of the acyclic quiver $Q$ attached to the GIM matrix $C$. From the resulting $2$-periodic triangulated categories, we build the corresponding Ringel--Hall Lie algebras, and establish a surjective Lie algebra homomorphism from each elliptic Lie algebra to its integral Ringel--Hall counterpart. This map is conjectured to be injective, and its injectivity on real root spaces is confirmed.