Double Poisson brackets and involutive representation spaces

TL;DR

This paper extends Van den Bergh's double Poisson brackets to involutive representation spaces, creating new Poisson structures on subspaces of representation spaces defined by symmetry conditions.

Contribution

It introduces an analog of Van den Bergh's construction that produces Poisson structures on involutive representation spaces, a new class of subspaces with symmetry constraints.

Findings

Constructs Poisson structures on involutive representation spaces.

Establishes a framework linking double Poisson brackets to symmetric subspaces.

Provides examples of involutive representation spaces with induced Poisson structures.

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 $\operatorname{Rep}(A,d)$ 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.