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On a Poisson homogeneous space of bilinear forms with a Poisson-Lie action. (English. Russian original) Zbl 1411.53070

Russ. Math. Surv. 72, No. 6, 1109-1156 (2017); translation from Usp. Mat. Nauk 72, No. 6, 139-190 (2017).
This paper studies the Poisson geometry of a natural \(\mathrm{GL}_{N}\left( \mathbb{C}\right)\)-action on bilinear forms on \(\mathbb{C}^{N}\). More precisely, \(A\mapsto BAB^{T}\) for \(A,B\in \mathrm{GL}_{N}\left( \mathbb{C}\right)\) defines an action of \(\mathrm{GL}_{N}\left( \mathbb{C}\right)\) on itself, and hence a transformation group groupoid \(\mathfrak{G}:=\mathrm{GL}_{N}\left( \mathbb{C} \right) \ltimes \mathrm{GL}_{N}\left( \mathbb{C}\right)\). Let \(\mathcal{A}\) be the set of all upper triangular matrices \(\mathbb{A}\in \mathrm{GL}_{N}\left( \mathbb{C}\right)\) with diagonal entries 1. Restricting the groupoid \(\mathfrak{G}\) to the subset \(\mathcal{A}\) of its unit space \(\mathrm{GL}_{N}\left( \mathbb{C}\right)\) yields a Lie groupoid \(\mathfrak{G}|_{\mathcal{A}}\) consisting of elements of \(\mathfrak{G}\) with both of their sources and targets lying in \(\mathcal{A}\). Associated with the Lie groupoid \(\mathfrak{G}|_{\mathcal{A}}\) is a Lie algebroid \(\mathfrak{g}\), which can be identified with the Lie algebroid \(T^{\ast}\mathcal{A}\).
In this paper, the authors study the more general case of \(\mathcal{A} _{n,m}\subset \mathrm{GL}_{N}\left( \mathbb{C}\right)\) consisting of block upper triangular matrices \(A=\left( A_{I,J}\right) _{1\leq I,J\leq n}\) with \(A_{I,J}\in M_{m}\left( \mathbb{C}\right)\) and \(\det\left( A_{I,I}\right) =1\) for the diagonal blocks \(A_{I,I}\), where \(N=nm\). A Lie algebroid is constructed and integrated into a Lie groupoid \(\mathcal{M}_{n,m}\).
The quadratic Poisson structures on \(\mathrm{GL}_{N}\times\mathcal{A}\) extending the standard Poisson-Lie structure on \(\mathrm{GL}_{N}\) while making \(\mathbb{A}\mapsto B\mathbb{A}B^{T}\) a Poisson action on \(\mathcal{A}\) are classified into three types (i), (ii), and (iii). Furthermore \(\mathcal{A}\) can be identified via the map \(\left(B,C\right) \mapsto BC^{T}\) with the Poisson symmetric space \(\left( \mathrm{GL}_{N}\times \mathrm{GL}_{N}\right) /H\) for the fixed point space \(H\) of the involutive anti-Poisson automorphism \(\left( B,C\right) \mapsto\left( C^{-T},B^{-T}\right)\) on \(\mathrm{GL}_{N}\times \mathrm{GL}_{N}\). The quadratic Poisson structures on \(\mathrm{GL}_{N}\times \mathrm{GL}_{N}\times\mathcal{A}\) making \(\mathbb{A}\mapsto B\mathbb{A}C^{T}\) a Poisson action on \(\mathcal{A}\) with \(B,C\in \mathrm{GL}_{N}\) are similarly classified. More generally, Poisson structures on the space of chains \(\left( A,B_{1},\dots,B_{j}\right)\) (and also \(\left( A,B_{1},C_{1},\dots,B_{j},C_{j}\right)\)) compatible with the groupoid multiple product are determined. The central elements for all these Poisson structures are classified.
It is shown that the procedure of Dirac reduction to the space \(\mathcal{A} _{n,m}\) fails for the Poisson structures of type (i), but works for those of type (ii) or (iii) explicitly in the case of \(\mathcal{A}_{n,1}\). Furthermore a quantum affine version of the Poisson algebra for Poisson structures of type (ii) on \(\mathrm{GL}_{N}\times\mathcal{A}\) is derived.

MSC:

53D17 Poisson manifolds; Poisson groupoids and algebroids
16T25 Yang-Baxter equations
20L05 Groupoids (i.e. small categories in which all morphisms are isomorphisms)
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