Lie--Poisson pencils related to semisimple Lie algebras: towards classification

TL;DR

This paper advances the classification of compatible Lie brackets on semisimple Lie algebras by fixing operators preserving root decompositions, leading to known and new examples, and conjecturing the completeness of these classifications.

Contribution

It introduces a novel approach fixing operators to classify compatible Lie brackets on semisimple Lie algebras, revealing new examples and organizing known ones.

Findings

Classification into two disjoint classes based on symmetry properties.

Recovery of known examples within each class.

Identification of new examples and conjecture of list completeness.

Abstract

Let $\mathfrak{g}$ be a vector space and $[,],[,]'$ be a pair of Lie brackets on $\mathfrak{g}$. By definition they are compatible if $[,]+[,]'$ is again a Lie bracket. Such pairs play important role in bihamiltonian and $r$-matrix formalisms in the theory of integrable systems. We propose an approach to a long standing problem of classification of such pairs in the case when one of them, say $[,]$, is semisimple. It is known that any such pair is determined by a linear operator on $(\mathfrak{g},[,])$, which is defined up to adding a derivation. We propose a special fixing of this operator to get rid of this ambiguity and consider the operators preserving the root decomposition with respect to a Cartan subalgebra. The classification leads to two disjoint classes of pairs depending on the symmetry properties of the corresponding operator with respect to the Killing form. Within each class we recover known examples and obtain new ones. We present a list of examples in each case and conjecture the completeness of these lists.