arXiv 0904.09242009-04math.RAmath.RT

可解李A-代数

Solvable Lie A-algebras

David A. Towers

摘要：

域$F$上的有限维李代数$L$称为$A$ -代数，如果它的所有幂零子代数都是阿贝尔的.这类似于$A$ -群的概念：一个有限群，其所有的Sylow子群都是阿贝尔群。这些群体在1940年代由菲利普·霍尔（Philip Hall）首次研究，至今仍在研究。关于$A$ -代数的知识较少，尽管它们已经被许多作者研究和使用。本文的目的是得到可解李$A$ -代数的结构的更详细的结果。它表明，他们分裂在其派生系列的每一项。这导致$L$分解为$L = A_{n} \dot{+} A_{n-1} \dot{+} ... \dot{+} A_0$，其中$A_i$是$L$的阿贝尔子代数，并且对于每个$0 \leq i \leq n$，$L^{(i)} = A_{n} \dot{+} A_{n-1} \dot{+} ... \dot{+} A_{i}$。它表明，理想的$L$很好地涉及到这个分解：如果$K$是一个理想的$L$然后$K = (K \cap A_n) \dot{+} (K \cap A_{n-1}) \dot{+} ... \dot{+} (K \cap A_0)$。当$L^2$是幂零的时，我们可以确定极大幂零子代数的位置：如果$U$是$L$的极大幂零子代数，则$U = (U \cap L^2) \oplus (U \cap C)$，其中$C$是$L$的Cartan子代数.若$L$有唯一极小理想$W$则$N = Z_L(W)$ .此外，如果$L$是强可解的，则$L$的极大幂零子代数是$L^2$和$L$的Cartan子代数（即与$L^2$互补的子代数）。给出了这类代数是$A$ -代数的充要条件.最后，当底层域是代数闭的时，给出了更详细的结构结果。

英文摘要：

A finite-dimensional Lie algebra $L$ over a field $F$ is called an $A$-algebra if all of its nilpotent subalgebras are abelian. This is analogous to the concept of an $A$-group: a finite group with the property that all of its Sylow subgroups are abelian. These groups were first studied in the 1940s by Philip Hall, and are still studied today. Rather less is known about $A$-algebras, though they have been studied and used by a number of authors. The purpose of this paper is to obtain more detailed results on the structure of solvable Lie $A$-algebras. It is shown that they split over each term in their derived series. This leads to a decomposition of $L$ as $L = A_{n} \dot{+} A_{n-1} \dot{+} ... \dot{+} A_0$ where $A_i$ is an abelian subalgebra of $L$ and $L^{(i)} = A_{n} \dot{+} A_{n-1} \dot{+} ... \dot{+} A_{i}$ for each $0 \leq i \leq n$. It is shown that the ideals of $L$ relate nicely to this decomposition: if $K$ is an ideal of $L$ then $K = (K \cap A_n) \dot{+} (K \cap A_{n-1}) \dot{+} ... \dot{+} (K \cap A_0)$. When $L^2$ is nilpotent we can locate the position of the maximal nilpotent subalgebras: if $U$ is a maximal nilpotent subalgebra of $L$ then $U = (U \cap L^2) \oplus (U \cap C)$ where $C$ is a Cartan subalgebra of $L$. If $L$ has a unique minimal ideal $W$ then $N = Z_L(W)$. If, in addition, $L$ is strongly solvable the maximal nilpotent subalgebras of $L$ are $L^2$ and the Cartan subalgebras of $L$ (that is, the subalgebras that are complementary to $L^2$.) Necessary and sufficient conditions are given for such an algebra to be an $A$-algebra. Finally, more detailed structure results are given when the underlying field is algebraically closed.

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摘要：

英文摘要：

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