Domestic canonical algebras and simple Lie algebras

TL;DR

This paper constructs simple complex Lie algebras of simply-laced Dynkin types as quotients of degenerate composition Lie algebras associated with domestic canonical algebras, providing explicit descriptions and bases for root spaces.

Contribution

It explicitly realizes simple Lie algebras as quotients of Hall algebra-based Lie algebras from domestic canonical algebras, with detailed descriptions of the ideals and bases.

Findings

Realization of simple Lie algebras as quotients of degenerate composition Lie algebras.

Explicit form of the ideal defining the quotient.

Root space bases given by indecomposable modules with computable dimension vectors.

Abstract

For each simply-laced Dynkin graph $\Delta$ we realize the simple complex Lie algebra of type $\Delta$ as a quotient algebra of the complex degenerate composition Lie algebra $L(A)_{1}^{\mathbb{C}}$ of a domestic canonical algebra $A$ of type $\Delta$ by some ideal $I$ of $L(A)_{1}^{\mathbb{C}}$ that is defined via the Hall algebra of $A$, and give an explicit form of $I$. Moreover, we show that each root space of $L(A)_{1}^{\mathbb{C}}/I$ has a basis given by the coset of an indecomposable $A$-module $M$ with root easily computed by the dimension vector of $M$.