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

Let $\mathscr A$ be the space of bilinear forms on $\mathbb C^N$ with defining matrices $\mathbb A$ 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 $\mathbb A\mapsto B\mathbb AB^{\mathrm{T}}$ of the $\mathrm{GL}_N$ Poisson–Lie group on $\mathscr A$. A classification is given of all possible quadratic brackets on $(B,\mathbb A)\in \mathrm{GL}_N\times \mathscr A$ preserving the Poisson property of the action, thus endowing $\mathscr A$ with the structure of a Poisson homogeneous space. Besides the product Poisson structure on $\mathrm{GL}_N\times \mathscr A$, there are two other (mutually dual) structures, which (unlike the product Poisson structure) 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 $(B,C,\mathbb A)\in \mathrm{GL}_N\times \mathrm{GL}_N\times \mathscr A$ with the Poisson action $\mathbb A\mapsto B\mathbb AC^{\mathrm{T}}$, and it is shown that $\mathscr A$ then acquires the structure of a Poisson symmetric space. Generalisations to chains of transformations and to the quantum and quantum affine algebras are investigated, as well as the relations between constructions of Poisson symmetric spaces and the Poisson groupoid.
Bibliography: 30 titles.