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 L_n, Q_n, R_n, W_n and also for all filiform Lie algebras of dimension at most eight. For the latter the Poisson center of these Lie algebras is determined.

中文翻译：

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李代数的最大阿贝尔维数，Rentschler的性质和Milovanov的猜想

如果具有维数为c（L）的有限维李代数L满足伦茨勒的性质，前提是它允许维数至少为c（L）− 1的阿贝尔李子代数H。我们研究这种新性质在各种李代数中的出现，例如不可解，可解，幂等，变态和丝状李代数。在某些温和条件下，H产生对称代数S（L）的完整Poisson可交换子代数。利用这一点，我们证明Milovanov猜想对类型为L _n，Q _n的丝状李代数成立，R _n，W _n以及所有维数最多为8的丝状李代数。对于后者，确定了这些李代数的泊松中心。