Solvable Lie A-algebras
By D. Towers. 2009
248 citations recorded by the source corpus.
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.
Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/9e97d1d78a2b0d31bcfbb850c6d797d42b21df66
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Citations in the corpus
- Classification of solvable Leibniz algebras with null-filiform nilradical (2012, result): “Thanks to Mal’cev’s results [11] the study of solvable Lie algebras is reduced to the study of nilpotent ones.”
- Lattices in almost abelian Lie groups with locally conformal Kähler or symplectic structures (2016, unknown): “However, there are two classes of Lie groups for which a criterion exists: for nilpotent Lie groups, there is the well known Malcev’s theorem [29], while for almost abelian Lie groups there is a characterization by Bock ([7], see Proposition 2.”