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 \({\text {Rep}}(A,d)\) 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.