On the structure of the necklace Lie algebra

TL;DR

This paper systematically studies the Lie algebra structure of the necklace Lie algebra associated with a free algebra in two variables, revealing its module decomposition, center, and Poisson structure, with connections to matrix trace rings and Poisson orders.

Contribution

It provides a detailed decomposition of the necklace Lie algebra into modules, describes its center and Poisson structure, and links double Poisson algebras to Poisson orders, offering new insights into their algebraic structures.

Findings

Decomposition of necklace Lie algebra into highest weight modules

Identification of the center linked to trace rings of matrices

Description of symplectic leaves as coadjoint orbits

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.