2016/01/01 by Balázs Elek, Jiang-Hua Lu, Elek, Balazs +1 · 2 citations
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Advanced Algebra and Geometry #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.1601.00047
Let G be a connected complex semi-simple Lie group, and let Z\bf u be an n-dimensional Bott-Samelson variety of G, where \bf u is any sequence of simple reflections in the Weyl group of G. We study the Poisson structure πn on Z\bf u defined by a standard multiplicative Poisson structure π\rm st on G. We explicitly express πn on each of the 2n affine coordinate charts, one for every subexpression of \bf u, in terms of the root strings and the structure constants of the Lie algebra of G. We show that the restriction of πn to each affine coordinate chart gives rise to a Poisson structure on the polynomial algebra ℂ[z1, …, zn] which is an \it iterated Poisson Ore extension of ℂ compatible with a rational action by a maximal torus of G. For canonically chosen π\rm st, we show that the induced Poisson structure on ℂ[z1, …, zn] for every affine coordinate chart is in fact defined over \mathbb Z, thus giving rise to an iterated Poisson Ore extension of any field \bf k of arbitrary characteristic. The special case of πn on the affine chart corresponding to the full subexpression of \bf u yields an explicit formula for the standard Poisson structures on \it generalized Bruhat cells in Bott-Samelson coordinates. The paper establishes the foundation on generalized Bruhat cells and sets up the stage for their applications, some of which are discussed in the Introduction of the paper.