Double Poisson Brackets on Low-Dimensional Algebras

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.