Triangulated categories and Kac-Moody algebras
The positive part of a finite dimensional semisimple complex Lie algebra can be constructed from the category of representations of an associated Dynkin quiver, by Ringel's Hall algebra method. Earlier work of \textit{L. Peng} and \textit{J. Xiao} [see J. Algebra 198, No. 1, 19-56 (1997; Zbl 0893.16007) and CMS Conf. Proc. 18, 539-554 (1996; Zbl 0870.16009)] used the derived category of quiver representations modulo the square of the shift functor to construct the full Lie algebra. The present paper follows a much more general approach; it starts with a triangulated category (over a finite field) such that \(T^2=1\) and constructs a Lie algebra from it. The most complicated part is checking the Jacobi identity, which involves a lot of counting of morphisms and triangles. Specializing to representation categories of quivers, one then passes to a generic form in order to replace the finite ground field by the complex numbers. In this way, the authors manage to construct all symmetrizable Kac-Moody Lie algebras.
scientific article; zbMATH DE number 1462437