Derived subalgebras of centralisers and finite -algebras
Derived subalgebras of centralisers and finite -algebras
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中心化子和有限代数的导出子代数
DOI：
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发表时间：
2013
期刊：
影响因子：
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通讯作者：
L. Topley
中科院分区：
文献类型：
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作者：
A. Premet;L. Topley
设g = Lie（G）是特征为0的代数闭域上的单代数群G的李代数。设e是g的幂零元，g_e = Lie（G_e），其中G_e表示e在G中的稳定子.对于g经典，给出了[g_e，g_e]在g_e中余维数的一个显式组合公式，并利用它确定了g中有限W-代数U（g，e）的最大交换商U（g，e）^{ab}同构于多项式代数的那些e.结果证明，当且仅当e位于g的唯一片中时，这种情况才会发生。本文称具有这种性质的幂零元为非奇异元。本文引用Izosimov最近的一个猜想，证明了g中的幂零元e是非奇异的当且仅当g中包含e的片的几何同分元S/G的最大维数与g中[g_e，g_e]的余维数一致，并用分拆来描述所有非奇异幂零元.我们还证明了对经典李代数g中的任何幂零元e，Specm U（g，e）^{ab}的闭子集同构于仿射空间。类似的例外李代数的这些结果也得到和应用的理论的原始理想。
Let g = Lie(G) be the Lie algebra of a simple algebraic group G over an algebraically closed field of characteristic 0. Let e be a nilpotent element of g and let g_e = Lie(G_e) where G_e stands for the stabiliser of e in G. For g classical, we give an explicit combinatorial formula for the codimension of [g_e, g_e] in g_e and use it to determine those e in g for which the largest commutative quotient U(g,e)^{ab} of the finite W-algebra U(g,e) is isomorphic to a polynomial algebra. It turns out that this happens if and only if e lies in a unique sheet of g. The nilpotent elements with this property are called non-singular in the paper. Confirming a recent conjecture of Izosimov we prove that a nilpotent element e in g is non-singular if and only if the maximal dimension of the geometric quotients S/G, where S is a sheet of g containing e, coincides with the codimension of [g_e,g_e] in g_e and describe all non-singular nilpotent elements in terms of partitions. We also show that for any nilpotent element e in a classical Lie algebra g the closed subset of Specm U(g,e)^{ab} consisting of all points fixed by the natural action of the component group of G_e is isomorphic to an affine space. Analogues of these results for exceptional Lie algebras are also obtained and applications to the theory of primitive ideals are given.