We show that any CPA-structure (commutative post-Lie algebra structure) on a perfect Lie algebra is trivial. Furthermore we give a general decomposition of inner CPA-structures, and classify all ...CPA-structures on complete Lie algebras. As a special case we obtain the CPA-structures of parabolic subalgebras of semisimple Lie algebras.
We study disemisimple Lie algebras, i.e., Lie algebras which can be written as a vector space sum of two semisimple subalgebras. We show that a Lie algebra g is disemisimple if and only if its ...solvable radical coincides with its nilradical and is a prehomogeneous s-module for a Levi subalgebra s of g. We use the classification of prehomogeneous s-modules for simple Lie algebras s given by Vinberg to show that the solvable radical of a disemisimple Lie algebra with simple Levi subalgebra is abelian. We extend this result to disemisimple Lie algebras having no simple quotients of type A.
We study post-Lie algebra structures on pairs of Lie algebras (g,n), which describe simply transitive nil-affine actions of Lie groups. We prove existence results for such structures depending on the ...interplay of the algebraic structures of g and n. We consider the classes of simple, semisimple, reductive, perfect, solvable, nilpotent, abelian and unimodular Lie algebras. Furthermore we consider commutative post-Lie algebra structures on perfect Lie algebras. Using Lie algebra cohomology we can classify such structures in several cases. We also study commutative structures on low-dimensional Lie algebras and on nilpotent Lie algebras.
A sandwich in thin lie algebras Mattarei, Sandro
Proceedings of the Edinburgh Mathematical Society,
02/2022, Letnik:
65, Številka:
1
Journal Article
Recenzirano
Odprti dostop
A thin Lie algebra is a Lie algebra $L$, graded over the positive integers, with its first homogeneous component $L_1$ of dimension two and generating $L$, and such that each non-zero ideal of $L$ ...lies between consecutive terms of its lower central series. All homogeneous components of a thin Lie algebra have dimension one or two, and the two-dimensional components are called diamonds. Suppose the second diamond of $L$ (that is, the next diamond past $L_1$) occurs in degree $k$. We prove that if $k>5$, then $Lyy=0$ for some non-zero element $y$ of $L_1$. In characteristic different from two this means $y$ is a sandwich element of $L$. We discuss the relevance of this fact in connection with an important theorem of Premet on sandwich elements in modular Lie algebras.
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