143 research outputs found
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
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 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
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