$W$ -代数与双哈密顿约化、Drinfeld-Sokolov约化和Dirac约化的等价性
$W$-algebras and the equivalence of bihamiltonian, Drinfeld-Sokolov and Dirac reductions
摘要：
证明了与简单李代数中的幂零轨道相关联的经典$W$ -代数可以通过预化为双哈密顿量、Drinfeld-Sokolov约化或Dirac约化来构造.我们得出结论：经典的$W$ -代数只依赖于幂零轨道，而不依赖于好的分次或迷向子空间的选择.此外，利用这个结果，我们再次证明了幂零轨道的横向Poisson结构是多项式的，我们更好地阐明了经典和有限$W$ -代数之间的关系。
英文摘要：
We prove that the classical $W$-algebra associated to a nilpotent orbit in a simple Lie-algebra can be constructed by preforming bihamiltonian, Drinfeld-Sokolov or Dirac reductions. We conclude that the classical $W$-algebra depends only on the nilpotent orbit but not on the choice of a good grading or an isotropic subspace. In addition, using this result we prove again that the transverse Poisson structure to a nilpotent orbit is polynomial and we better clarify the relation between classical and finite $W$-algebras.
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