Double Poisson brackets and involutive representation spaces

Let $A$ be a finitely generated associative algebra, in general noncommutative. A double Poisson bracket on $A$ in the sense of Van den Bergh arXiv:math/0410528 is a bilinear map $\{\{-,-\}\}$ from $A\times A$ to $A^{\otimes 2}$, subject to certain conditions. Van den Bergh showed that any such bracket $\{\{-,-\}\}$ induces a Poisson structure on the space $\mathrm{Rep}(A,N)$ of $N$-dimensional representations of the algebra $A$ for any $N$. We propose an analog of Van den Bergh's construction, which produces Poisson structures on certain subspaces of the representation spaces $\mathrm{Rep}(A,N)$. We call these subspaces the involutive representation spaces. They arise by imposing an additional symmetry condition on $\mathrm{Rep}(A,N)$ -- just as the classical groups from the series B, C, D are obtained from the general linear groups (series A) as fixed point sets of involutive automorphisms. The talk is based on a joint paper with Grigori Olshanski arXiv:2310.01086.