Transposed Poisson structures on the Lie algebra of upper triangular matrices

Abstract

We describe transposed Poisson structures on the upper triangular matrix Lie algebra $T_n(F)$, $n>1$, over a field $F$ of characteristic zero. We prove that, for $n>2$, any such structure is either of Poisson type or the orthogonal sum of a fixed non-Poisson structure with a structure of Poisson type, and for $n=2$, there is one more class of transposed Poisson structures on $T_n(F)$. We also show that, up to isomorphism, the full matrix Lie algebra $M_n(F)$ admits only one non-trivial transposed Poisson structure, and it is of Poisson type.