Double Poisson brackets and involutive representation spaces

Abstract

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 ${\text {Rep}}(A,d)$ of d-dimensional representations of the algebra A. A double Poisson bracket on A in the sense of Van den Bergh (Trans Am Math Soc 360:5711–5799, 2008) 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 ${\text {Rep}}(A,d)$. We call these subspaces the involutive representation spaces. They arise by imposing an additional symmetry condition on ${\text {Rep}}(A,d)$—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.