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Deformations of T-log-symplectic log-canonical Poisson structures and symmetric Poisson CGL extensions

Jiang-Hua Lu, Mykola Matviichuk

TL;DR

The article develops a framework for ${oldsymbol{ ext T}}$-invariant Poisson deformations of ${oldsymbol{ ext T}}$-log-symplectic log-canonical structures on ${oldsymbol{ ext C}}^n$, proving unobstructedness for first-order deformations under suitable weight conditions. It then shows that, for ${oldsymbol{ ext T}}$-action data, one can canonically deform these structures to ${oldsymbol{ ext T}}$-invariant algebraic Poisson structures that are strongly symmetric Poisson ${oldsymbol{ ext CGL}}$ extensions in the sense of Goodearl-Yakimov, with a precise combinatorial control via weight sets ${oldsymbol{ ext S}}({oldsymbol{ ext pi}}_0)$. In particular, for symmetrizable generalized Cartan matrices, the construction yields families ${oldsymbol{ ext pi}}^{({f i})}(c)$ that coincide with standard Poisson structures on Bott-Samelson and generalized Schubert cells in finite type, thereby connecting Poisson geometry, Lie theory, and cluster structures. The work provides a unified elementary approach to obtaining symmetric Poisson CGL extensions from Cartan data and action data and establishes a bridge to cluster algebras on key Lie-theoretic varieties. These results generalize to Kac-Moody settings in ongoing directions and illuminate smoothings of Poisson divisors via Pfaffians in the ${oldsymbol{ ext T}}$-setting.

Abstract

For a complex algebraic torus $\mathbb{T}$, we study $\mathbb{T}$-invariant Poisson deformations of $\mathbb{T}$-log-symplectic log-canonical Poisson structures on $\mathbb{C}^n$. We show that, under mild assumptions, every $\mathbb{T}$-invariant first-order deformation with no $(\mathbb{C}^\times)^n$-invariant component is unobstructed. As an application, we prove that a special class of $\mathbb{T}$-log-symplectic log-canonical Poisson structures on $\mathbb{C}^n$, namely those defined by the so-called $\mathbb{T}$-action data, can be canonically deformed to $\mathbb{T}$-invariant algebraic Poisson structures on $\mathbb{C}^n$ that are (strongly) symmetric Poisson CGL extensions (of $\mathbb{C})$ in the sense of Goodearl-Yakimov. In particular, we construct (strongly) symmetric CGL extensions from any sequence of simple roots associated to any symmetrizable generalized Cartan matrix $A$. When $A$ is of finite type, our construction recovers the standard Poisson structures on Bott-Samelson cells and on generalized Schubert cells, which are closely related to the (standard) cluster algebra structures on such cells.

Deformations of T-log-symplectic log-canonical Poisson structures and symmetric Poisson CGL extensions

TL;DR

The article develops a framework for -invariant Poisson deformations of -log-symplectic log-canonical structures on , proving unobstructedness for first-order deformations under suitable weight conditions. It then shows that, for -action data, one can canonically deform these structures to -invariant algebraic Poisson structures that are strongly symmetric Poisson extensions in the sense of Goodearl-Yakimov, with a precise combinatorial control via weight sets . In particular, for symmetrizable generalized Cartan matrices, the construction yields families that coincide with standard Poisson structures on Bott-Samelson and generalized Schubert cells in finite type, thereby connecting Poisson geometry, Lie theory, and cluster structures. The work provides a unified elementary approach to obtaining symmetric Poisson CGL extensions from Cartan data and action data and establishes a bridge to cluster algebras on key Lie-theoretic varieties. These results generalize to Kac-Moody settings in ongoing directions and illuminate smoothings of Poisson divisors via Pfaffians in the -setting.

Abstract

For a complex algebraic torus , we study -invariant Poisson deformations of -log-symplectic log-canonical Poisson structures on . We show that, under mild assumptions, every -invariant first-order deformation with no -invariant component is unobstructed. As an application, we prove that a special class of -log-symplectic log-canonical Poisson structures on , namely those defined by the so-called -action data, can be canonically deformed to -invariant algebraic Poisson structures on that are (strongly) symmetric Poisson CGL extensions (of in the sense of Goodearl-Yakimov. In particular, we construct (strongly) symmetric CGL extensions from any sequence of simple roots associated to any symmetrizable generalized Cartan matrix . When is of finite type, our construction recovers the standard Poisson structures on Bott-Samelson cells and on generalized Schubert cells, which are closely related to the (standard) cluster algebra structures on such cells.

Paper Structure

This paper contains 16 sections, 31 theorems, 189 equations.

Table of Contents

  1. Introduction and main results
  2. Introduction
  3. Deformations of ${\mathbb{T}}$-log-symplectic log-canonical Poisson structures
  4. Applications to symmetric Poisson CGL extensions
  5. Notation
  6. Acknowledgments
  7. Deformations of ${\mathbb{T}}$-log-symplectic log-canonical Poisson structures
  8. Poisson cohomology of log-canonical Poisson structures
  9. Poisson cohomology of ${\mathbb{T}}$-log-symplectic log-canonical Poisson structures
  10. Formal deformations of ${\mathbb{T}}$-log-symplectic log-canonical Poisson structures
  11. Algebraic deformations of ${\mathbb{T}}$-log-symplectic log-canonical Poisson structures
  12. Symmetric Poisson CGL extensions from ${\mathbb{T}}$-action data
  13. ${\mathbb{T}}$-log-symplectic log-canonical Poisson structures from ${\mathbb{T}}$-action data
  14. Symmetric Poisson CGL extensions from ${\mathbb{T}}$-action data
  15. Symmetric Poisson CGL extensions from generalized Cartan matrices
  16. ...and 1 more sections

Key Result

Lemma 1

For ${\bf u}, {\bf w} \in ({\mathbb Z}_{\geq -1})^n$ and $J, K \subset [1, n]$ with $J_{\bf w} \subset J$ and $J_{\bf u} \subset K$, one has

Theorems & Definitions (84)

  • Lemma 1
  • Lemma 2
  • proof
  • Remark 1
  • Definition 1
  • Remark 2
  • Lemma 3
  • proof
  • Lemma 4
  • Lemma 5
  • ...and 74 more