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Towards a categorical construction of Lie algebras

This is an introduction to the program which we call “towards a categorical construction of Lie Algebras”. That is, from the data of a system of 4 integers W := (a, b, c;h), called a regular system of weights, satisfying an arithmetic condition, we want to construct a certain generalization gW of a simple Lie algebra. Precisely, to a weight system, we first associate a surface with a singular point. Then, using the geometry of the singularity, a triangulated category is attached. Finally, we want to read Lie theoretic data from the category and to construct the algebra gW . The program is still in its early stages, and, in the present paper, we are mainly concerned with some categorical aspects of the program, and then ask questions on the possible constructions of Lie algebras. The organization of the paper is as follows. In §1-9, we start by recalling the classical relations of simple or simply elliptic singularities with simple or elliptic Lie algebras, respectively, as the prototype of relations between singularities and Lie algebras. This part is rather