Nilpotent Lie Algebras Research Articles

A Deleting Derivations Algorithm for Quantum Nilpotent Algebras at Roots of Unity

A Deleting Derivations Algorithm for Quantum Nilpotent Algebras at Roots of Unity

The algebraic classification of low-dimensional nilpotent n-Lie algebras

In this paper, we study the low-dimensional nilpotent [Formula: see text]-Lie algebras and classify nilpotent [Formula: see text]-Lie algebras of dimension [Formula: see text] and nilpotent [Formula: see text]-Lie algebras of class two with dimension [Formula: see text].

Strong Kähler with torsion solvable lie algebras with codimension 2 nilradical

AbstractIn this paper, we study strong Kähler with torsion (SKT) and generalized Kähler structures on solvable Lie algebras with (not necessarily abelian) codimension 2 nilradical. We treat separately the case of ‐invariant nilradical and non‐‐invariant nilradical. A classification of such SKT Lie algebras in dimension 6 is provided. In particular, we give a general construction to extend SKT nilpotent Lie algebras to SKT solvable Lie algebras of higher dimension, and we construct new examples of SKT and generalized Kähler compact solvmanifolds.

The classification of nilpotent Bol algebras

In this paper, we generalize the classical method used to classify nilpotent low-dimensional Lie algebras to serve for nilpotent Bol algebras. The algebraic classification of the nilpotent Bol algebras up to dimension four is given.

Model-theoretic properties of nilpotent groups and Lie algebras

We give a systematic study of the model theory of generic nilpotent groups and Lie algebras. We show that the Fraïssé limit of 2-nilpotent groups of exponent p studied by Baudisch is 2-dependent and NSOP1. We prove that the class of c-nilpotent Lie algebras over an arbitrary field, in a language with predicates for a Lazard series, is closed under free amalgamation. We show that for 2<c, the generic c-nilpotent Lie algebra over Fp is strictly NSOP4 and c-dependent. Via the Lazard correspondence, we obtain the same result for c-nilpotent groups of exponent p, for an odd prime p>c.

Malcev-like binary Lie algebras of dimension 5

We study anti-commutative algebras, which are extensions of a one-dimensional algebra by a four-dimensional nilpotent Lie algebra and at the same time extensions of the two-dimensional non-Abelian Lie algebra by an abelian algebra. These algebras, just like the five-dimensional solvable Malcev algebras, have a flag of subalgebras and can therefore be considered their closest relatives. We characterize binary Lie algebras in this class, called Malcev-like algebras, playing interesting role in non-associative Lie theory. We find normal forms of their multiplications, and determine their isomorphism classes.

A Construction of Einstein Solvmanifolds not Based on Nilsolitons

AbstractWe construct indefinite Einstein solvmanifolds that are standard, but not of pseudo-Iwasawa type. Thus, the underlying Lie algebras take the form $$\mathfrak {g}\rtimes _D\mathbb {R}$$ g ⋊ D R , where $$\mathfrak {g}$$ g is a nilpotent Lie algebra and D is a nonsymmetric derivation. Considering nonsymmetric derivations has the consequence that $$\mathfrak {g}$$ g is not a nilsoliton, but satisfies a more general condition. Our construction is based on the notion of nondiagonal triple on a nice diagram. We present an algorithm to classify nondiagonal triples and the associated Einstein metrics. With the use of a computer, we obtain all solutions up to dimension 5, and all solutions in dimension $$\le 9$$ ≤ 9 that satisfy an additional technical restriction. By comparing curvatures, we show that the Einstein solvmanifolds of dimension $$\le 5$$ ≤ 5 that we obtain by our construction are not isometric to a standard extension of a nilsoliton.

Structure and Representations for Electrical Lie Algebra of Type D5

In this paper, we prove that the electrical Lie algebra [Formula: see text] is isomorphic to the semidirect product of [Formula: see text] and a 2-step nilpotent Lie algebra. Furthermore, we classify the irreducible highest weight modules for [Formula: see text].

A characterization of Anosov rational forms in nilpotent Lie algebras associated to graphs

A characterization of Anosov rational forms in nilpotent Lie algebras associated to graphs

Skew-Adjoint Maps and Quadratic Lie Algebras

The procedure of double extension of vector spaces endowed with non-degenerate bilinear forms allows us to introduce the class of generalized K\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {K}$$\\end{document}-oscillator algebras over an arbitrary field K\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {K}$$\\end{document}. Starting from basic structural properties and the canonical forms of skew-adjoint endomorphisms, we will proceed to classify the subclass of quadratic nilpotent algebras and characterize those algebras with quadratic dimension 2. This will enable us to recover the classification of real oscillator algebras, a.k.a Lorentzian algebras, given by Medina et al. (Ann Sci École Norm Sup 18:553–561, 1985).

On decompositions and transitive actions of nilpotent Lie groups

The article considers decompositions of nilpotent Lie algebras and nilpotent Lie groups, and connections between them. Also, descriptions of irreducible transitive actions of nilpotent Lie groups on the plane and on three-dimensional space are given.

Classification of flat Lorentzian nilpotent Lie algebras

AbstractWe give a complete classification of flat Lorentzian nilpotent Lie algebras, this is to say of pseudo‐Euclidean Lie algebras associated to nilpotent Lie groups endowed with a left‐invariant Lorentzian metric of vanishing curvature. We prove that every such a Lie algebra is a direct sum of an indecomposable flat Lorentzian Lie algebra and an abelian Euclidean summand and show that, if denotes the ‐dimensional Heisenberg Lie algebra, then the only non‐abelian Lie algebras admitting flat Lorentzian metrics which are indecomposable are and the semidirect products and , defined by some particular derivations . In all those cases we also find the equivalence classes of flat Lorentzian products.

Ad-invariant metrics on nonnice nilpotent Lie algebras

We proved in previous work that all real nilpotent Lie algebras of dimension up to [Formula: see text] carrying an ad-invariant metric are nice, i.e. they admit a nice basis in the sense of Lauret et al. In this paper, we show by constructing explicit examples that nonnice irreducible nilpotent Lie algebras admitting an ad-invariant metric exist for every dimension greater than [Formula: see text] and every nilpotency step greater than [Formula: see text]. In the way of doing so, we introduce a method to construct Lie algebras with ad-invariant metrics called the single extension, as a parallel to the well-known double extension procedure.

Isotopisms of nilpotent Leibniz algebras and Lie racks

In this paper we study the isotopism classes of two-step nilpotent algebras. We show that every nilpotent Leibniz algebra g with dim [ g , g ] = 1 is isotopic to the Heisenberg Lie algebra or to the Heisenberg algebra l 2 n + 1 J 1 , where J 1 is the n × n Jordan block of eigenvalue 1. We also prove that two such algebras are isotopic if and only if the Lie racks integrating them are isotopic. This gives the classification of Lie racks whose tangent space at the unit element is a nilpotent Leibniz algebra with one-dimensional commutator ideal. Eventually, we introduce new isotopism invariants for Leibniz algebras and Lie racks.

Investigation of ID-derivations of finite-dimensional nilpotent Lie algebras

Let [Formula: see text] be a Lie algebra and let [Formula: see text] be the set of all derivations of [Formula: see text]. Then the derivation [Formula: see text] of [Formula: see text] is called an ID-derivation if [Formula: see text] for all [Formula: see text]. The set of all ID-derivations is denoted by [Formula: see text]. Let [Formula: see text] be the set of all central derivations and let [Formula: see text] and [Formula: see text] be, respectively, the set of all center derivations and ID-derivations that map [Formula: see text] to zero. In this paper, we verify relations between [Formula: see text] and [Formula: see text] with [Formula: see text] and [Formula: see text]. Also, we prove that if [Formula: see text] is a nilpotent Lie algebra of class [Formula: see text], then [Formula: see text] is a nilpotent Lie algebra of class [Formula: see text].

Rings of finite Morley rank without the canonical base property

We present numerous natural algebraic examples without the so-called Canonical Base Property (CBP). We prove that every commutative unitary ring of finite Morley rank without finite-index proper ideals satisfies the CBP if and only if it is a field, a ring of positive characteristic or a finite direct product of these. In addition, we construct a CM-trivial commutative local ring with a finite residue field without the CBP. Furthermore, we also show that finite-dimensional non-associative algebras over an algebraically closed field of characteristic [Formula: see text] give rise to triangular rings without the CBP. This also applies to Baudisch’s [Formula: see text]-step nilpotent Lie algebras, which yields the existence of a [Formula: see text]-step nilpotent group of finite Morley rank whose theory, in the pure language of groups, is CM-trivial and does not satisfy the CBP.

Shintani descent for standard supercharacters of algebra groups

Let [Formula: see text] be a finite-dimensional nilpotent algebra over a finite field [Formula: see text] with q elements, and let [Formula: see text]. On the other hand, let [Formula: see text] denote the algebraic closure of [Formula: see text], and let [Formula: see text]. Then [Formula: see text] is an algebraic group over [Formula: see text] equipped with an [Formula: see text]-rational structure given by the usual Frobenius map [Formula: see text], and [Formula: see text] can be regarded as the fixed point subgroup [Formula: see text]. For every [Formula: see text], the nth power [Formula: see text] is also a Frobenius map, and [Formula: see text] identifies with [Formula: see text]. The Frobenius map restricts to a group automorphism [Formula: see text], and hence it acts on the set of irreducible characters of [Formula: see text]. Shintani descent provides a method to compare F-invariant irreducible characters of [Formula: see text] and irreducible characters of [Formula: see text]. In this paper, we show that it also provides a uniform way of studying supercharacters of [Formula: see text] for [Formula: see text]. These groups form an inductive system with respect to the inclusion maps [Formula: see text] whenever [Formula: see text], and this fact allows us to study all supercharacter theories simultaneously, to establish connections between them, and to relate them to the algebraic group G. Indeed, we show that Shintani descent permits the definition of a certain “superdual algebra” which encodes information about the supercharacters of [Formula: see text] for [Formula: see text].

Real non-degenerate two-step nilpotent Lie algebras of dimension eight

We classify the non-degenerate two-step nilpotent Lie algebras of dimension 8 over the field of real numbers, using known results over complex numbers. We write explicit structure constants for these real Lie algebras.

More on characteristic polynomials of Lie algebras

In recent years, the notion of characteristic polynomials of representations of Lie algebras has been widely studied. This paper provides more properties of these characteristic polynomials. For simple Lie algebras, we characterize the linearization of characteristic polynomials. Additionally, we characterize nilpotent Lie algebras via characteristic polynomials of the adjoint representation.

Bounds for the dimension of the Schur multiplier of finite dimensional nilpotent Lie algebras

Let L be an n-dimensional nilpotent Lie algebra of nilpotency class c≥2 over an arbitrary field F with dim⁡γ2(L)=m and dim(Z(L)Z(L)∩γ2(L))=t, where Z(L) and γ2(L) denote the center ideal and the derived subalgebra of L, respectively. The purpose of this work is to obtain a new bound on the dimension of the Schur multiplier for L, which is given in terms of n,m,c, and t. In the virtue of an old bound due to Niroomand and Russo, namely 12(n−m−1)(n+m−2)+1, we find a finite dimensional nilpotent Lie algebra L of nilpotency class 3 over a field F of characteristic 3 that attains this bound. We then present a complete list of all finite dimensional nilpotent Lie algebras of nilpotency class at least 2 that reach Niroomand–Russo's Formula. Finally, we give the structure of all finite dimensional nilpotent Lie algebras that attain some recent generalizations of Niroomand–Russo's Formula.