Drinfeld-Sokolov双哈密顿结构的Frobenius流形和中心不变量

$W$ -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

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。

Algebraic classical $W$-algebras and Frobenius manifolds

We consider Drinfeld-Sokolov bihamiltonian structure associated to a distinguished nilpotent elements of semisimple type and the space of common equilibrium points defined by its leading term. On this space, we construct a local bihamiltonian structure which form an exact Poisson pencil, defines an algebraic classical $W$-algebra, admits a dispersionless limit, and its leading term defines an algebraic Frobenius manifold. This leads to a uniform construction of algebraic Frobenius manifolds corresponding to regular cuspidal conjugacy classes in irreducible Weyl groups.