Double Poisson brackets and involutive representation spaces

Letters in Mathematical Physics. 2024. Vol. 114. No. 1. Article 33.

Olshanski G., Safonkin N.

Let k be an algebraically closed field of characteristic 0 and A be a finitely generated
associative k-algebra, in general noncommutative. One assigns to A a sequence
of commutative k-algebras O(A, d), d = 1, 2, 3, . . . , where O(A, d) is the coordinate
ring of the space 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 × A to A⊗A, subject
to certain conditions. Van den Bergh showed that any such bracket {{−, −}} induces
Poisson structures on all algebras 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 Rep(A, d).We call these subspaces the involutive
representation spaces. They arise by imposing an additional symmetry condition
on 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.

Keywords: Double Poisson brackets, Poisson algebras, Involutive representation spaces, Twisted Van den Bergh construction, Classical groups