[Paper Review] On a Poisson space of bilinear forms with a Poisson Lie action

Leonid Chekhov, Marta Mazzocco|arXiv (Cornell University)|Apr 3, 2014

Advanced Algebra and Geometry15 references3 citations

TL;DR

This paper classifies all quadratic Poisson structures on $GL_N \times \mathcal{A}$, where $\mathcal{A}$ is the space of bilinear forms on $\mathbb{C}^N$, such that the natural $GL_N$-action $\mathbb{A} \mapsto B\mathbb{A}B^T$ is a Poisson action. It identifies three such structures—two dual to each other and one product structure—showing that only the dual structures allow Dirac reduction to block upper-triangular matrices, and generalizes the construction to Poisson symmetric spaces via $GL_N \times GL_N \times \mathcal{A}$ with action $\mathbb{A} \mapsto B\mathbb{A}C^T$. The key contribution is a systematic classification of compatible Poisson structures for such actions, with implications for Poisson groupoids and quantum algebras.

ABSTRACT

We consider the space of bilinear forms on a complex N-dimensional vector space endowed with the quadratic Poisson bracket studied in our previous paper arXiv:1012.5251. We classify all possible quadratic brackets on the set of pairs of matrices A and B with the property that the natural action of B on the defining matrix A of a bilinear form is a Poisson action of a Poisson-Lie group, thus endowing this space of bilinear forms with the structure of Poisson homogeneous space. Beside the product Poisson structure we find two more (dual to each other) structures for which (in contrast to the product Poisson structure) we can implement the reduction to the space of bilinear forms with block upper triangular defining matrices by Dirac procedure. We consider the generalisation of the above construction to triples and show that the space of bilinear forms then acquires the structure of Poisson symmetric space. We study also the generalisation to chains of transformations and to the quantum and quantum affine algebras and the relation between the construction of Poisson symmetric spaces and that of the Poisson groupoid.

Motivation & Objective

• To classify all quadratic Poisson brackets on $GL_N \times \mathcal{A}$ such that the $GL_N$-action $\mathbb{A} \mapsto B\mathbb{A}B^T$ is a Poisson action.
• To determine which of these structures allow Dirac reduction to the subspace of block upper-triangular matrices in $\mathcal{A}$.
• To generalize the construction to triples $(B, C, \mathbb{A}) \in GL_N \times GL_N \times \mathcal{A}$ with action $\mathbb{A} \mapsto B\mathbb{A}C^T$, showing $\mathcal{A}$ becomes a Poisson symmetric space.
• To explore connections with Poisson groupoids, quantum algebras, and quantum affine algebras, particularly in the context of $r$-matrix formalism and $GL_N$-Poisson-Lie group structure.

Proposed method

• Assumes the standard Lie-Poisson bracket on $GL_N$ via the classical trigonometric $r$-matrix $r_{12} = 2\sum_{i>j} E_{ij} \otimes E_{ji} + \sum_i E_{ii} \otimes E_{ii}$.
• Derives the required cross-bracket $\{B \otimes \mathbb{A}\}$ as $B Q_{12} \mathbb{A} + B \mathbb{A} Q_{12}^{t_2}$, with $Q_{12}$ constrained to three specific choices: $0$, $-r_{12}^{t_2}$, or $r_{12}^{t_1}$.
• Proves that only the two non-product structures ($Q_{12} = \pm r_{12}^{t_2}$) are dual to each other and admit Dirac reduction to the block upper-triangular subspace $\mathcal{A}_{n,m}$.
• Uses the $r$-matrix formalism to express the Poisson bracket on $\mathcal{A}$ as $\{\mathbb{A}_1, \mathbb{A}_2\} = r_{12}(\mathbb{A}_1 \otimes \mathbb{A}_2) - (\mathbb{A}_1 \otimes \mathbb{A}_2)r_{12} + \mathbb{A}_1 r_{12}^{t_1} \mathbb{A}_2 - \mathbb{A}_2 r_{12}^{t_1} \mathbb{A}_1$, ensuring Jacobi identity via the classical Yang-Baxter equation.
• Applies the Dirac procedure to reduce the full space to subspaces with block upper-triangular $\mathbb{A}$, showing that only the dual structures preserve the Poisson structure under such constraints.
• Generalizes the framework to chains of transformations and discusses relations to Poisson groupoids and quantum algebras, particularly in the context of symplectic groupoid constructions and Bondal's symplectic forms.

Experimental results

Research questions

• RQ1 Which Poisson structures on $GL_N \times \mathcal{A}$ make the $GL_N$-action $\mathbb{A} \mapsto B\mathbb{A}B^T$ a Poisson action?
• RQ2 Can Dirac reduction be implemented to the subspace of block upper-triangular matrices $\mathbb{A}$, and if so, under which Poisson structures?
• RQ3 How do the three possible choices for the cross-bracket $\{B \otimes \mathbb{A}\}$—$Q_{12} = 0$, $-r_{12}^{t_2}$, or $r_{12}^{t_1}$—relate to each other, and which allow consistent reduction?
• RQ4 What is the generalization of this construction to actions $\mathbb{A} \mapsto B\mathbb{A}C^T$ with two $GL_N$ factors, and does $\mathcal{A}$ then become a Poisson symmetric space?
• RQ5 How does this framework relate to Poisson groupoids, symplectic groupoids, and quantum algebras, particularly in light of Bondal’s symplectic structure?

Key findings

• The paper classifies exactly three possible Poisson structures on $GL_N \times \mathcal{A}$ for which the $GL_N$-action $\mathbb{A} \mapsto B\mathbb{A}B^T$ is Poisson, corresponding to $Q_{12} = 0$, $-r_{12}^{t_2}$, and $r_{12}^{t_1}$.
• The product structure ($Q_{12} = 0$) does not allow Dirac reduction to the block upper-triangular subspace $\mathcal{A}_{n,m}$, while the two dual structures ($Q_{12} = \pm r_{12}^{t_2}$) do allow such reduction.
• The two dual structures are related by a canonical duality transformation, and both preserve the Poisson structure under the Dirac procedure on the constraint surface of block upper-triangular matrices.
• The generalization to $GL_N \times GL_N \times \mathcal{A}$ with action $\mathbb{A} \mapsto B\mathbb{A}C^T$ endows $\mathcal{A}$ with the structure of a Poisson symmetric space.
• The construction is compatible with the classical $r$-matrix formalism and satisfies the classical Yang-Baxter equation, ensuring Jacobi identity for the Poisson brackets.
• The paper conjectures that the symplectic structure on $({\mathbb{F}}, B)$ pairs generalizes Bondal’s symplectic form in the upper-triangular case, though the dynamics become trivial due to variable separation in the symplectic groupoid approach.