Domestic canonical algebras and simple Lie algebras

Details Domestic canonical algebras and simple Lie algebras Domestic canonical algebras and simple Lie algebras

2007-05-07

arxiv.org/abs/0705.0942v4

Mathematics

Representation Theory

43 pages, 5 figures, revised version

Scientific paper

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 $M$.

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