Study of hereditary categories and related topics. (R-9227)

In non-commutative algebraic geometry, one often studies abelian and triangulated categories with properties similar to those of the categories of coherent sheaves on smooth proper algebraic varieties, or to their derived categories. In this way, a category of global dimension one, called a hereditary category, can be interpreted as a category of coherent sheaves on a non-commutative curve. Examples of such categories are given by the category of representations of a quiver, or the category of coherent sheaves on a smooth projective (commutative) curve. Since the algebraic geometry of curves is relatively simple, it seems reasonable to assume that the classification of hereditary categories (satisfying some additional geometric conditions) should also be feasible. My research project concerns this classification, and the study of the new examples that occur to complete these classifications. Hereditary categories are a prototype of many interesting phenomena in representation theory and algebraic geometry, and have links to different areas, such as root systems, quantum groups, and noncrossing partitions. By studying these phenomena and links for the aforementioned new hereditary categories, we obtain new objects of interest, such as new quantum groups or the concept of triangulations of cyclically ordered sets.

Keywords:Derived and triangulated categories, Hereditary categories, Noncommutative geometry, Quantum groups, representation theory

Disciplines:Algebra