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 paper, we define a N=2 multi-component supersymmetric B type 2KP(MS2BKP) hierarchy and its additional symmetries. These additional flows form a B type multi SW1+∞ Lie algebra. Under a ...reduction, we derive a N=2 multi-component supersymmetric D type Drinfeld–Sokolov hierarchy which has a multi super Virasoro algebraic structure.
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.
Abstract
We study nilpotent Lie algebras endowed with a complex structure and a quadratic structure which is pseudo-Hermitian for the given complex structure. We propose several methods to construct ...such Lie algebras and describe a method of double extension by planes to get an inductive description of all of them. As an application, we give a complete classification of nilpotent quadratic Lie algebras where the metric is Lorentz-Hermitian and we fully classify all nilpotent pseudo-Hermitian quadratic Lie algebras up to dimension 8 and their inequivalent pseudo-Hermitian metrics.
Let A be the path algebra of a Dynkin quiver Q over a finite field, and P be the category of projective A-modules. Denote by C1(P) the category of 1-cyclic complexes over P, and n˜+ the vector space ...spanned by the isomorphism classes of indecomposable and non-acyclic objects in C1(P). In this paper, we prove the existence of Hall polynomials in C1(P), and then establish a relationship between the Hall numbers for indecomposable objects therein and those for A-modules. Using Hall polynomials evaluated at 1, we define a Lie bracket in n˜+ by the commutators of degenerate Hall multiplication. The resulting Hall Lie algebras provide a broad class of nilpotent Lie algebras. For example, if Q is bipartite, n˜+ is isomorphic to the nilpotent part of the corresponding semisimple Lie algebra; if Q is the linearly oriented quiver of type An, n˜+ is isomorphic to the free 2-step nilpotent Lie algebra with n-generators. Furthermore, we give a description of the root systems of different n˜+. We also characterize the Lie algebras n˜+ by generators and relations. When Q is of type A, the relations are exactly the defining relations. As a byproduct, we construct an orthogonal exceptional pair satisfying the minimal Horseshoe lemma for each sincere non-projective indecomposable A-module.
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