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

For a complex algebraic torus $\mathbb{T}$, we study $\mathbb{T}$-invariant Poisson deformations of a $\mathbb{T}$-log-symplectic log-canonical Poisson structure $\pi_0$ on $\mathbb{C}^n$. We show that every $\mathbb{T}$-invariant first-order deformation of $\pi_0$ with linearly independent $(\mathbb{C}^\times)^n$-weights is unobstructed. For a special class of $\pi_0$ defined by the so-called symmetric $\mathbb{T}$-action data, we show that $\pi_0$ 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 $\pi_0$ and the second $\mathbb{T}$-invariant Poisson cohomology of $\pi_0$. 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 $\pi$ with log-canonical term $\pi_0$, we present an explicit formula expressing the initial mutation matrix in the Goodearl-Yakimov theory on cluster algebras associated to $\pi$ in terms of the $(\mathbb{C}^\times)^n$-weights of the second $\mathbb{T}$-invariant Poisson cohomology of $\pi_0$.