On a Poisson homogeneous space of bilinear forms with a Poisson–Lie action

Let A be the space of bilinear forms on C^N with defining matrices A endowed with a quadratic Poisson structure of reflection equation type. The paper begins with a short description of previous studies of the structure, and then this structure is extended to systems of bilinear forms whose dynamics is governed by the natural action A ﰀ→ BAB^T of the GL_N Poisson–Lie group on A . A classification is given of all possible quadratic brackets on (B, A) ∈ GL_N ×A preserving the Poisson property of the action, thus endowing A with the structure of a Poisson homogeneous space. Besides the product Poisson structure on GL_N × A , there are two other (mutually dual) structures, which (unlike the product Poisson struc- ture) admit reductions by the Dirac procedure to a space of bilinear forms with block upper triangular defining matrices. Further generalisations of this construction are considered, to triples (B, C, A) ∈ GL_N × GL_N × A with the Poisson action A ﰀ→ BAC^T, and it is shown that A then acquires the structure of a Poisson symmetric space. Generalisations to chains of transformations and to the quantum and quantum affine algebras are inves- tigated, as well as the relations between constructions of Poisson symmetric spaces and the Poisson groupoid.