李代数的最大交换维数、Rentschler性质和Milovanov猜想
The maximal abelian dimension of a Lie algebra, Rentschler's property and Milovanov's conjecture
摘要：
一个具有幻数c（L）的有限维李代数L称为满足Rentschler性质，如果它允许一个维数至少为c（L）- 1的交换李子代数H.我们研究了这一新性质在各种李代数，如不可解、可解、幂零、亚交换和丝状李代数中的出现。在适当的条件下，H生成对称代数S（L）的一个完备Poisson交换子代数.利用这一点，我们证明了Milovanov猜想对Ln，Qn，Rn，Wn型的丝状李代数以及所有维数不超过8的丝状李代数都成立。对于后者，确定了这些李代数的Poisson中心。
英文摘要：
A finite dimensional Lie algebra L with magic number c(L) is said to satisfy Rentschler's property if it admits an abelian Lie subalgebra H of dimension at least c(L) - 1. We study the occurrence of this new property in various Lie algebras, such as nonsolvable, solvable, nilpotent, metabelian and filiform Lie algebras. Under some mild condition H gives rise to a complete Poisson commutative subalgebra of the symmetric algebra S(L). Using this, we show that Milovanov's conjecture holds for the filiform Lie algebras of type Ln, Qn, Rn, Wn and also for all filiform Lie algebras of dimension at most eight. For the latter the Poisson center of these Lie algebras is determined.
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