  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.
