Deformations of T-log-symplectic log-canonical Poisson structures and symmetric Poisson CGL extensions
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866911624108965888 |
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| author | Lu, Jiang-Hua Matviichuk, Mykola |
| author_facet | Lu, Jiang-Hua Matviichuk, Mykola |
| contents | For a complex algebraic torus $\mathbb{T}$, we study $\mathbb{T}$-invariant Poisson deformations of a $\mathbb{T}$-log-symplectic log-canonical Poisson structure $π_0$ on $\mathbb{C}^n$. We show that every $\mathbb{T}$-invariant first-order deformation of $π_0$ with linearly independent $(\mathbb{C}^\times)^n$-weights is unobstructed. For a special class of $π_0$ defined by the so-called symmetric $\mathbb{T}$-action data, we show that $π_0$ can be canonically deformed to symmetric $\mathbb{T}$-Poisson CGL extensions (of $\mathbb{C}$) as defined by K. Goodearl and M. Yakimov. As a consequence, we classify all symmetric $\mathbb{T}$-Poisson CGL extensions in terms of their log-canonical terms $π_0$ and the second $\mathbb{T}$-invariant Poisson cohomology of $π_0$. We further characterize, among all symmetric Poisson CGL extensions, those of Cartan type, i.e., those associated to sequences of simple roots in the root systems of symmetrizable generalized Cartan matrices. In particular, we prove that the standard Poisson structures on Bott-Samelson cells and generalized Schubert cells for semi-simple complex Lie groups are the (uniquely determined) maximal normalized admissible deformations of their log-canonical terms. Finally, for any symmetric $\mathbb{T}$-Poisson CGL extension $π$ with log-canonical term $π_0$, we present an explicit formula expressing the initial mutation matrix in the Goodearl-Yakimov theory on cluster algebras associated to $π$ in terms of the $(\mathbb{C}^\times)^n$-weights of the second $\mathbb{T}$-invariant Poisson cohomology of $π_0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_05644 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Deformations of T-log-symplectic log-canonical Poisson structures and symmetric Poisson CGL extensions Lu, Jiang-Hua Matviichuk, Mykola Symplectic Geometry Rings and Algebras Representation Theory 17B63 (Primary) 53D17, 14M15, 13F60 (Secondary) For a complex algebraic torus $\mathbb{T}$, we study $\mathbb{T}$-invariant Poisson deformations of a $\mathbb{T}$-log-symplectic log-canonical Poisson structure $π_0$ on $\mathbb{C}^n$. We show that every $\mathbb{T}$-invariant first-order deformation of $π_0$ with linearly independent $(\mathbb{C}^\times)^n$-weights is unobstructed. For a special class of $π_0$ defined by the so-called symmetric $\mathbb{T}$-action data, we show that $π_0$ can be canonically deformed to symmetric $\mathbb{T}$-Poisson CGL extensions (of $\mathbb{C}$) as defined by K. Goodearl and M. Yakimov. As a consequence, we classify all symmetric $\mathbb{T}$-Poisson CGL extensions in terms of their log-canonical terms $π_0$ and the second $\mathbb{T}$-invariant Poisson cohomology of $π_0$. We further characterize, among all symmetric Poisson CGL extensions, those of Cartan type, i.e., those associated to sequences of simple roots in the root systems of symmetrizable generalized Cartan matrices. In particular, we prove that the standard Poisson structures on Bott-Samelson cells and generalized Schubert cells for semi-simple complex Lie groups are the (uniquely determined) maximal normalized admissible deformations of their log-canonical terms. Finally, for any symmetric $\mathbb{T}$-Poisson CGL extension $π$ with log-canonical term $π_0$, we present an explicit formula expressing the initial mutation matrix in the Goodearl-Yakimov theory on cluster algebras associated to $π$ in terms of the $(\mathbb{C}^\times)^n$-weights of the second $\mathbb{T}$-invariant Poisson cohomology of $π_0$. |
| title | Deformations of T-log-symplectic log-canonical Poisson structures and symmetric Poisson CGL extensions |
| topic | Symplectic Geometry Rings and Algebras Representation Theory 17B63 (Primary) 53D17, 14M15, 13F60 (Secondary) |
| url | https://arxiv.org/abs/2503.05644 |