Infinity Algebras and the Homology of Graph Complexes

An A-infinity algebra is a generalization of a associative algebra, and an L-infinity algebra is a generalization of a Lie algebra. In this paper, we show that an L-infinity algebra with an invariant inner product determines a cycle in the homology of the complex of metric ordinary graphs. Since the cyclic cohomology of a Lie algebra with an invariant inner product determines infinitesimal deformations of the Lie algebra into an L-infinity algebra with an invariant inner product, this construction shows that a cyclic cocycle of a Lie algebra determines a cycle in the homology of the graph complex. In this paper a simple proof of the corresponding result for A-infinity algebras, which was proved in a different manner in an earlier paper, is given.Comment: 14 pages, amslatex document, 4 figure

Ribbon Graphs, Quadratic Differentials on Riemann Surfaces, and Algebraic Curves Defined over Qˉ\bar Q

It is well known that there is a bijective correspondence between metric ribbon graphs and compact Riemann surfaces with meromorphic Strebel differentials. In this article, it is proved that Grothendieck's correspondence between dessins d'enfants and Belyi morphisms is a special case of this correspondence. For a metric ribbon graph with edge length 1, an algebraic curve over $\bar Q$ and a Strebel differential on it is constructed. It is also shown that the critical trajectories of the measured foliation that is determined by the Strebel differential recover the original metric ribbon graph. Conversely, for every Belyi morphism, a unique Strebel differential is constructed such that the critical leaves of the measured foliation it determines form a metric ribbon graph of edge length 1, which coincides with the corresponding dessin d'enfant.Comment: Higher resolution figures available at http://math.ucdavis.edu/~mulase