Double Poisson Brackets on Low-Dimensional Algebras

G. I. Sharygin^abc, A. Hernández Rodríguez<sup>d

a</sup> Lomonosov Moscow State University, Leninskie Gory 1, Moscow, 119991 Russia
^b National Research Centre “Kurchatov Institute”, pl. Akad. Kurchatova 1, Moscow, 123182 Russia
^c Moscow Institute of Physics and Technology (National Research University), Institutskii per. 9, Dolgoprudnyi, Moscow oblast, 141701 Russia
^d National Autonomous University of Mexico (UNAM), Av. Universidad 3000, Col. UNAM, Coyoacán, 04510 Mexico city, Mexico

Abstract: We describe double Poisson brackets in the sense of M. Van den Bergh on certain finite-dimensional algebras. In particular, we prove that all possible double Poisson brackets on matrix algebras are “inner,” i.e., given by some commutators in bimodules. As a corollary of this result, we see that all possible double Poisson brackets in any finite-dimensional semisimple algebras over algebraically closed fields are also given by inner derivations. We also describe all double Poisson brackets on the algebra of $2\times 2$ upper triangular matrices. We further discuss Poisson structures induced from the double Poisson brackets in its representation spaces of rank $2$ and $3$. In the appendix, we describe modified double Poisson brackets (in the sense of S. Arthamonov) on this algebra.

Keywords: double Poisson brackets, representation spaces, Poisson structures.

Received: May 5, 2025
Revised: June 9, 2025
Accepted: July 17, 2025

DOI: 10.4213/tm4487

 English version: Proceedings of the Steklov Institute of Mathematics, 2025, 330, 359–374

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