Double Poisson brackets and involutive representation spaces

Let $\Bbbk$ be an algebraically closed field of characteristic $0$ and $A$ be a finitely generated associative $\Bbbk$-algebra, in general noncommutative. One assigns to $A$ a sequence of commutative $\Bbbk$-algebras $\mathcal{O}(A,d)$, $d=1,2,3,\dots$, where $\mathcal{O}(A,d)$ is the coordinate ring of the space $\operatorname{Rep}(A,d)$ of $d$-dimensional representations of the algebra $A$. A double Poisson bracket on $A$ in the sense of Van den Bergh [Trans. Amer. Math. Soc. (2008); 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 Poisson structures on all algebras $\mathcal{O}(A,d)$. We propose an analog of Van den Bergh's construction, which produces Poisson structures on the coordinate rings of certain subspaces of the representation spaces $\operatorname{Rep}(A,d)$. We call these subspaces the involutive representation spaces. They arise by imposing an additional symmetry condition on $\operatorname{Rep}(A,d)$ -- 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.