Double Transposed Poisson Algebras

We introduce double transposed Poisson algebras, a noncommutative analogue of the transposed Poisson algebras of Bai, Bai, Guo and Wu that is compatible with the Kontsevich–Rosenberg principle. We first consider a simplified version which we call id-adapted double transposed Poisson algebras and then explore the general definition. We prove that every such structure on a unital associative algebra $\mathbb{A}$ is governed by a single derivation $\mathbb{A}\to \mathbb{A}\otimes \mathrm{S}(\mathbb{A}\mathrm{/}[\mathbb{A},\mathbb{A}])$. Furthermore, this induces a ${\mathrm{GL}}_N$-equivariant transposed Poisson structure on each representation algebra ${\mathbb{A}}_N=\mathbb{k}[{\mathrm{Rep}}_N(\mathbb{A})]$. We also define $H_0$-transposed Poisson structures, the transposed counterpart of Crawley-Boevey’s $H_0$-Poisson structures, and use the trace map to obtain a transposed Poisson structure on the ring of ${\mathrm{GL}}_N$-invariants ${\mathbb{A}}_N^{{\mathrm{GL}}_N}$.