Poisson Centralizer of the Trace

TL;DR

This paper identifies the Poisson centralizer of the trace in the coordinate ring of SL_n, showing it coincides with the invariant subalgebra under the adjoint action, linking Poisson geometry and invariant theory.

Contribution

It explicitly determines the Poisson centralizer of the trace in SL_n's coordinate ring, revealing its equivalence to the invariant subalgebra under the adjoint action.

Findings

Poisson centralizer of the trace equals the invariant subalgebra

Maximal Poisson-commutative subalgebra identified

Connection established between Poisson geometry and invariants

Abstract

The Poisson centralizer of the trace element is determined in the coordinate ring of SL_n endowed with the Poisson structure obtained as the semiclassical limit of its quantized coordinate ring. It turns out that this maximal Poisson-commutative subalgebra coincides with the subalgebra of invariants with respect to the adjoint action.