Algebra Seminar 

01/06/2005, 11:00 — 12:00 — Room P3.10, Mathematics Building
Paulo Lima-Filho, Texas A&M University

On the holonomy Lie algebra of graphic arrangements

If $X$ is the complement of a projective hypersurface, then the nilpotent completion of its fundamental group is isomorphic to the nilpotent completion of the holonomy Lie algebra of $X$, as shown by Kohno. In the particular case where $X$ is the complement of a hyperplane arrangement $A$, the ranks ${\phi }_k$ of the lower central series quotients of ${\pi }_1(X)$ are well-known in only two very special cases: if the arrangement $A$ is hypersolvable (a linear slice of an arrangement which admits a sequence of linear fibrations), or if the the holonomy Lie algebra decomposes in degree greater or equal to $2$ as a direct product of local components. In this talk we show how to use the holonomy Lie algebra to obtain an explicit combinatorial formula for the ranks ${\phi }_k$, whenever $A$ is a graphic arrangement. This formula generalizes Kohno's result for braid arrangements, and provides the first instance of a lower central series formula for a large class of arrangements which are not decomposable or fiber-type.