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We realize simple complex Lie algebras as quotient Lie algebras defined via Hall algebras of tame hereditary algebras. Let A be a finite-dimensional hereditary algebra over a finite field k with q elements, and consider the free abelian group HðAÞ with basis the isoclasses of finite A-modules. Then by Ringel (13) HðAÞ turns out to be an associative ring with identity, called the Hall algebra of A, with respect to the multiplication whose structure constants are given by the numbers of filtrations of modules with factors isomorphic to modules that are multiplied (see section 1 for details). The free abelian subgroup LðAÞ of HðAÞ with basis the isoclasses of finite indecomposable A-modules becomes a Lie subalgebra modulo q  1 whose Lie bracket is given by the commutator. It would be interesting to realize all types of simple (complex) Lie algebras using this Lie bracket given by the commutator of the Hall multiplication. Along this line, Ringel (14) realized the positive part of the simple Lie algebra gðDÞ for each Dynkin type D. Further Peng and Xiao (10) realized the all types of simple Lie algebras by the so-called root categories of finite-dimensional representation-finite hereditary algebras. But the Lie bracket was not completely given by the above type, because the root category R provides only the positive and the negative parts. The Hall multiplication was used to define the Lie bracket only inside R, and when the bracket should not be closed in R the definition was changed. In (1) we succeeded to realize general linear algebras and special linear algebras (i.e. the simple Lie algebras of type An) by this Lie bracket defined on cyclic quiver algebras. In this realization also the Cartan subalgebra was naturally provided together with the positive and the negative parts. The purpose of this paper is to give a way how to realize all types of simple Lie algebras by the Lie bracket given by the Hall multiplication, in particular to give explicit realization of simple Lie algebras of type Dn. Here we use Hall algebras of tame hereditary algebras (a‰ne quiver algebras in the simply-laced case), which were well studied by Ringel (15) and Peng and Xiao (11). 1. Preliminaries.

Realization of simple Lie algebras via Hall algebras of tame hereditary algebras