12,000

We have over 12,000 students, from over 100 countries, within one of the safest campuses in the UK

93%

93% of Lancaster students go into work or further study within six months of graduating

Home > Research > Publications & Outputs > Solvable Lie A-algebras.
View graph of relations

« Back

Solvable Lie A-algebras.

Research output: Contribution to journalJournal article

Published

Journal publication date15/08/2011
JournalJournal of Algebra
Journal number1
Volume340
Number of pages12
Pages1-12
Original languageEnglish

Abstract

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. \par It is shown that they split over each term in their derived series. This leads to a decomposition of $L$ as $L = A_{n} \dot{+} A_{n-1} \dot{+} \ldots \dot{+} A_0$ where $A_i$ is an abelian subalgebra of $L$ and $L^{(i)} = A_{n} \dot{+} A_{n-1} \dot{+} \ldots \dot{+} A_{i}$ 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 $K = (K \cap A_n) \dot{+} (K \cap A_{n-1}) \dot{+} \ldots \dot{+} (K \cap A_0)$. 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 $L$. \par 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.

Bibliographic note

The final, definitive version of this article has been published in the Journal, Journal of Algebra, 340 (1), 2011, © ELSEVIER.

Related research outputs