Casimirs of the Goldman Lie algebra of a closed surface

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In 1986 Goldman introduced a Lie algebra structure on the linear span L of conjugacy classes of the fundamental group of a closed oriented surface. It is easy to see that the class e₁ of the trivial loop is a central element in L. We prove that any central element of L is a multiple of e₁ (a conjecture communicated to the author by Chas and Sullivan), and moreover that any central element of the Poisson algebra SL is a polynomial of e₁.