Goldman [2] and Turaev [4] found a Lie bialgebra structure on the vector space generated by non-trivial free homotopy classes of loops on an orientable surface. Chas [1] by the aid of the computer, found a negative answer to Turaev's question about the characterization of the classes with cobracket zero as multiples of simple classes, in every surface of negative Euler characteristic and positive genus. However, she left open Turaev's conjecture, namely if, for genus zero, every class with cobracket zero is a multiple of a simple class. The aim of this paper is to give a positive answer to this conjecture.
On Lie Bialgebras of Loops on Orientable Surfaces / LE DONNE, Attilio. - In: JOURNAL OF KNOT THEORY AND ITS RAMIFICATIONS. - ISSN 0218-2165. - STAMPA. - 17, no. 3:(2008), pp. 1-9. [10.1142/S0218216508006178]
On Lie Bialgebras of Loops on Orientable Surfaces
LE DONNE, Attilio
2008
Abstract
Goldman [2] and Turaev [4] found a Lie bialgebra structure on the vector space generated by non-trivial free homotopy classes of loops on an orientable surface. Chas [1] by the aid of the computer, found a negative answer to Turaev's question about the characterization of the classes with cobracket zero as multiples of simple classes, in every surface of negative Euler characteristic and positive genus. However, she left open Turaev's conjecture, namely if, for genus zero, every class with cobracket zero is a multiple of a simple class. The aim of this paper is to give a positive answer to this conjecture.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.