Realization of simple Lie algebras via Hall algebras of tame hereditary algebras.
Representation theory of quivers is closely related to finite dimensional complex simple Lie algebras or Kac-Moody algebras. In particular, C.~M.~Ringel has shown that in the Dynkin case the positive part of the quantized enveloping algebra is isomorphic to the Hall algebra of any corresponding quiver. J.~A.~Green has generalized this result to Kac-Moody algebras. Specialising the quantum parameter, one can realize in this way the positive part of Lie algebras via quiver representations. In order to realize the whole Lie algebras in terms of quiver representations, \textit{L.~Peng} and \textit{J.~Xiao} [Invent. Math. 140, No. 3, 563-603 (2000; Zbl 0966.16006), J. Algebra 198, No. 1, 19-56 (1997; Zbl 0893.16007)] used root categories, which are quotients of derived categories of quiver representations. This approach does not directly produce the simple Lie algebra; some technical modifications are needed in order to deal with the Cartan part. The paper under review follows a different approach to realize the whole finite dimensional simple Lie algebra completely in terms of representations of quivers, but paying the price of using affine diagrams. It is known that a simple Lie algebra of Dynkin type can be written as a quotient of the positive part of an affine Lie algebra, using a quotient map \(\varphi\) sending the last Chevalley generator to a lowest root vector. Identifying the positive part of the affine Lie algebra with a composition algebra of a quiver, the realization is obtained. The main result of the paper makes this realization explicit by precisely giving the relation generating the kernel of \(\varphi\). The cases of types \(A\) and \(C\) are then worked out in detail.
scientific article; zbMATH DE number 2111279