On Lie algebra decompositions related to spherical homogeneous spaces
On Lie algebra decompositions related to spherical homogeneous spaces
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关于球齐次空间的李代数分解
DOI：
10.1007/bf03026538
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发表时间：
1993
影响因子：
0.6
通讯作者：
D. Akhiezer
中科院分区：
文献类型：
--
作者：
D. Akhiezer
设G是特征为零的代数闭域k上的连通的约化线性代数群。设H1和H2是G的两个球面子群。本文证明了对G的Zerkiki开子集中的所有g，有一个李代数分解g = h1 + Adg <$h2，其中a是环面的李代数，且dim a ≤ min（rankG/H1，rankG/H2）.作为应用，得到了域k（G/H1 × G/H2）G对G的对角作用的超越度的一个估计.如果k = α，且Ga是由反全纯对合σ：G→G定义的G的真实的形式，则对于球面子群H <$G，且对于G的Hausdorff开子集中的所有g，有分解g = ga + aAdg <$h，其中a是σ-不变环面的李代数且dim a ≤ rankG/H.
LetG be a connected, reductive, linear algebraic group over an algebraically closed fieldk of characteristik zero. LetH1 andH2 be two spherical subgroups ofG. It is shown that for allg in a Zariski open subset ofG one has a Lie algebra decomposition g = h1 + Adg ⋅ h2, where a is the Lie algebra of a torus and dim a ≤ min (rankG/H1,rankG/H2). As an application one obtains an estimate of the transcendence degree of the fieldk(G/H1 xG/H2)G for the diagonal action ofG. Ifk = ℂ andGa is a real form ofG defined by an antiholomorphic involution σ :G→G then for a spherical subgroup H ⊂ G and for allg in a Hausdorff open subset ofG one has a decomposition g = ga + a Adg ⋅ h, where a is the Lie algebra of σ-invariant torus and dim a ≤ rankG/H.