Relying on the classification of simple Lie algebras over algebraically closed fields of characteristic >3, we show that any finite-dimensional central simple 5-graded Lie algebra over a field k of ...characteristic ≠2,3 is a simple Lie algebra of Chevalley type, i.e. a central quotient of the Lie algebra of a simple algebraic k-group. As a consequence, we prove that all central simple structurable algebras and Kantor pairs over k arise from 5-gradings on simple Lie algebras of Chevalley type.
The index of a Lie algebra is an important algebraic invariant, but it is notoriously difficult to compute. However, for the suggestively-named seaweed algebras, the computation of the index can be ...reduced to a combinatorial formula based on the connected components of a “meander”: a planar graph associated with the algebra. Our index analysis on seaweed algebras requires only basic linear and abstract algebra. Indeed, the main goal of this article is to introduce a broader audience to seaweed algebras with minimal appeal to specialized language and notation from Lie theory. This said, we present several results that do not appear elsewhere and do appeal to more advanced language in the Introduction to provide added context.
Poisson cohomology of 3D Lie algebras Hoekstra, Douwe; Zeiser, Florian
Journal of geometry and physics,
September 2023, 2023-09-00, Letnik:
191
Journal Article
Recenzirano
Odprti dostop
We compute the Poisson cohomology associated with several three dimensional Lie algebras. Together with existing results and the classification of three dimensional Lie algebras, this provides the ...Poisson cohomology of all linear Poisson structures in dimension 3.
Commutative post-Lie algebra structures on Lie algebras, in short CPA structures, have been studied over fields of characteristic zero, in particular for real and complex numbers motivated by ...geometry. A perfect Lie algebra in characteristic zero only admits the trivial CPA-structure. In this article we study these structures over fields of characteristic p>0. We show that every perfect modular Lie algebra in characteristic p>2 having a solvable outer derivation algebra admits only the trivial CPA-structure. This involves a conjecture by Hans Zassenhaus, saying that the outer derivation algebra Out(g) of a simple modular Lie algebra g is solvable. We try to summarize the known results on the Zassenhaus conjecture and prove some new results using the classification of simple modular Lie algebras by Premet and Strade for algebraically closed fields of characteristic p>3. As a corollary we obtain that every central simple modular Lie algebra of characteristic p>3 admits only the trivial CPA-structure.
On Ricci negative derivations Gutiérrez, María Valeria
Advances in geometry,
04/2022, Letnik:
22, Številka:
2
Journal Article
Recenzirano
Odprti dostop
Given a nilpotent Lie algebra, we study the space of all diagonalizable derivations such that the corresponding one-dimensional solvable extension admits a left-invariant metric with negative Ricci ...curvature. Lauret and Will have conjectured that such a space coincides with an open and convex subset of derivations defined in terms of the moment map for the variety of nilpotent Lie algebras. We prove the validity of the conjecture in dimensions ≤ 5, as well as for Heisenberg Lie algebras and standard filiform Lie algebras.
In this study, we consider Lie algebras that admit para-Kähler and hyper-para-Kähler structures. We provide new characterizations of these Lie algebras and develop many methods for building large ...classes of examples. Previously, Bai considered para-Kähler Lie algebras as left symmetric bialgebras. We reconsider this viewpoint and make improvements in order to obtain some new results. The study of para-Kähler and hyper-para-Kähler is intimately linked to the study of left symmetric algebras, particularly those that admit invariant symplectic forms. In this study, we provide many new classes of left symmetric algebras and complete descriptions of all the associative algebras that admit an invariant symplectic form. We also describe all four-dimensional hyper-para-Kähler Lie algebras.
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