Elliptic Root System and Elliptic Artin Group

Abstract

The concepts of root system, Dynkin diagram and Weyl group were introduced by K. Saito to describe the Milnor lattices and the flat structures of semi-universal deformations for simply singularities [S], [SO], [SI], [S2]. Generators and relations of Weyl were studied in the context of Dynkin diagrams by K. Saito and T. Takebayashi [ST]. (This presentation of an Weyl group is a generalization of a Coxeter system. See Theorem 2.1). In their paper, they proposed the following problem: find generators and relations of elliptic Lie algebras, elliptic Hecke and elliptic Artin groups (the fundamental of the complements of the discriminant for simply singularities) in terms of the Dynkin diagrams, In [SY], applying R. Borcherd's construction of vertex algebras [Borl], [Bor2], K. Saito and D. Yoshii constructed the Lie algebras (which are isomorphic to the toroidal algebras [MEY]) for homogeneous Dynkin diagrams. In addition, they described the fundamental relations in terms of the generators attached to the Dynkin diagrams. These relations are a generalization of the Serre type relations (for other appraches cf. [BM], [Sll]). In this article, we shall give an answer to their problem for the case of Artin and Hecke algebras as an application of the twisted Picard-Lefschetz formula due to A. B. Givental [Gi]. As for the former groups, they have already been studied by H. van der Lek