Modified double brackets and a conjecture of S. Arthamonov

TL;DR

This paper introduces mixed double Poisson algebras, proving a conjecture by Arthamonov by constructing new modified double Poisson brackets that extend the existing theory.

Contribution

It defines and studies mixed double Poisson algebras, resolving Arthamonov's conjecture by constructing a new family of modified double Poisson brackets.

Findings

Resolved Arthamonov's conjecture for the second case.

Introduced and analyzed mixed double Poisson algebras.

Connected modified double Poisson brackets with Rota-Baxter operators.

Abstract

Around 20 years ago, M. Van den Bergh introduced double Poisson brackets as operations on associative algebras inducing Poisson brackets under the representation functor. Weaker versions of these operations, called modified double Poisson brackets, were later introduced by S. Arthamonov in order to induce a Poisson bracket on moduli spaces of representations of the corresponding associative algebras. Moreover, he defined two operations that he conjectured to be modified double Poisson brackets. The first case of this conjecture was recently proved by M. Goncharov and V. Gubarev motivated by the theory of Rota-Baxter operators of nonzero weight. We settle the conjecture by realising the second case as part of a new family of modified double Poisson brackets. These are obtained from mixed double Poisson algebras, a new class of algebraic structures that are introduced and studied in the present work.