Derive categories and infinite dimensional Lie allgebras associated with primitive forms

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The purpose of the program is the study is A. primitive forms, and B. thermodynamical limit functions for discrete groups.A. Towards categorical construction of primitive forms, 1. we constructed the highest weight integrable representations of elliptic Lie algebras, and 2. we determine the strongly exceptional collections of the categories of matrix factorizations for simply elliptic and 14 exceptional unimodular singularities (joint work with Takahashi, Kajiura and Oda). On the other hand, We started the construction of primitive forms for the function of type A and D with infinite critical points.B. We generalized the theory of the thermodynamical limit functions for a discrete group to any cancellative monoid. Then, the study of growth functions for cancellative monoids had big progress, and the inversion formula of the growth function is constructed in the monoid level. In particular, the monoid of integral square matrices shows a strong similarity with Artin monoids.