Bott–Samelson Varieties and Poisson Ore Extensions

Abstract

We show that associated with any $n$-dimensional Bott–Samelson variety of a complex semi-simple Lie group $G$, one has $2^n$ Poisson brackets on the polynomial algebra $A={\mathbb{C}}[z_1, \ldots , z_n]$, each an iterated Poisson Ore extension and one of them a symmetric Poisson Cauchon–Goodearl–Letzter (CGL) extension in the sense of Goodearl–Yakimov. We express the Poisson brackets in terms of root strings and structure constants of the Lie algebra of $G$. It follows that the coordinate rings of all generalized Bruhat cells have presentations as symmetric Poisson CGL extensions. The paper establishes the foundation on generalized Bruhat cells and sets the stage for their applications to integrable systems, cluster algebras, total positivity, and toric degenerations of Poisson varieties, some of which are discussed in the Introduction.

On the Standard Poisson Structure and a Frobenius Splitting of the Basic Affine Space

“…Therefore, (Z u , π u ) is a T -Poisson variety. The following theorem is proved in [7,Theorem 4.14] [7, Theorem 5.12] for the case k = C. Theorem 4.1. For any γ = (γ 1 , .…”

Generalized Bruhat Cells and Completeness of Hamiltonian Flows of Kogan-Zelevinsky Integrable Systems

“…It is easy to see that Z u is a (smooth and projective) Poisson submanifold of F n with respect to the Poisson structure π n , and (O u , π n ) is embedded in (Z u , π n ) as an open Poisson submanifold. The choice of the pinning in §1.4 gives rise [13,15] to the atlas…”

“…For each γ ∈ Υ u , it is shown in [14,15] that the Poisson structure π n is algebraic in the coordinate chart φ γ : C n → φ γ (C n ), and the Poisson brackets among the coordinate functions are expressed using root strings and structure constants of the Lie algebra g. Note, in particular, that when…”

“…4), the choice of a reduced word for each u j then gives rise to a parametrization of O (u 1 ,...,un) by C l(u 1 )+···+l(un) , where l is the length function on W , and we refer to the resulting coordinates on O (u 1 ,...,un) as Bott-Samelson coordinates (see Definition 2.7). Explicit formulas for the Poisson structure π n on O (u 1 ,...,un) in Bott-Samelson coordinates have been studied in [14,15]. By Bott-Samelson coordinates on a double Bruhat cell G u,v we mean the coordinates on G u,v obtained through the open Fomin-Zelevinsky embedding F u,v and the combination of coordinates on T and Bott-Samelson coordinates on O (v −1 ,u) .…”

“…We also point out that, based on the results in [14,15], B. Elek has written a computer program in the language of GAP that computes the Poisson structure π n on O (u 1 ,...,un) in the Bott-Samelson coordinates for any G and any sequence (u 1 , . .…”