Combinatorial Lie bialgebras of curves on surfaces
By Moira Chas. 2001
85 citations recorded by the source corpus.
Abstract Goldman (Invent. Math. 85(2) (1986) 263) and Turaev (Ann. Sci. Ecole Norm. Sup. (4) 24 (6) (1991) 635) found a Lie bialgebra structure on the vector space generated by non-trivial free homotopy classes of curves on a surface. When the surface has non-empty boundary, this vector space has a basis of cyclic reduced words in the generators of the fundamental group and their inverses. We give a combinatorial algorithm to compute this Lie bialgebra on this vector space of cyclic words. Using this presentation, we prove a variant of Goldman's result relating the bracket to disjointness of curve representatives when one of the classes is simple. We exhibit some examples we found by programming the algorithm which answer negatively Turaev's question about the characterization of simple curves in terms of the cobracket. Further computations suggest an alternative characterization of simple curves in terms of the bracket of a curve and its inverse. Turaev's question is still open in genus zero.
Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/a1e3fe49720f4666383a087322656b48f3a8c729
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Citations in the corpus
- The complex-symplectic geometry of SL(2,C)-characters over surfaces (2003, background): “nd fβ Poisson-commute. By arguments in [13] (based on a suggestion of S. Wolpert), the converse holds if one of αor βis simple. A purely topological proof of this fact has recently been given by Chas [5]. 2.3. Complex twist flows. Suppose that αis represented by a simple closed curve A. Then a holomo”