On the structure of the necklace Lie algebra

Abstract

In this note, we initiate the systematic study of the Lie algebra structure of the necklace Lie algebra n of a free algebra in 2d variables. We begin by giving a description of n as an sp(2d)-module. Specializing to d = 1, we decompose n into a direct sum of highest weight modules for sl_2, the coefficients of which are given by a closed formula. Next, we observe that n has a nontrivial center, which we link through the center C of the trace ring of couples of generic 2x2 matrices to the Poisson center of S(sl_2). The Lie algebra structure of n induces a Poisson structure on C, the symplectic leaves of which we are able to describe as coadjoint orbits for the Lie group of the semidirect product sl_2\rtimes h of sl_2 with the Heisenberg Lie algebra h. Finally, we provide a link between double Poisson algebras on one hand and Poisson orders on the other hand, showing that all trace rings of a double Poisson algebra are Poisson orders over their center.