XXII.—Some Nilpotent Algebras with Centres of Given Dimension

1961 ◽  
Vol 65 (4) ◽  
pp. 310-317
Author(s):  
E. W. Wallace
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Bilinear Form ◽  
Canonical Forms ◽  
Invariant Bilinear Form ◽  
Alternative Derivation ◽  
Direct Sum ◽  
Derived Algebras ◽  
Nilpotent Algebras

SYNOPSISAlgebras which are nilpotent and anti-commutative are studied. Canonical forms are found for all such algebras of dimension n whose centres have dimension n−r (r < 3), and characters are given which enable any two non-isomorphic algebras to be distinguished.A metrisable Lie algebra is a Lie algebra for which there is a non-singular, symmetric, adjoint-invariant bilinear form a(λ, μ), and such an algebra is reduced if its centre is contained in its derived algebra. The importance of the reduced algebras follows from the fact that every metrisable Lie algebra is the direct sum of a reduced metrisable Lie algebra and an abelian Lie algebra. Tsou (Thesis 1955) introduced metrisable Lie algebras, and obtained canonical forms for all real reduced metrisable Lie algebras whose derived algebras have dimension 3. We conclude this paper by providing an alternative derivation, two of the algebras being nilpotent.

scholarly journals Rings with involution whose symmetric elements are central

1980 ◽  
Vol 3 (2) ◽  
pp. 247-253
Author(s):  
Taw Pin Lim
Keyword(s):  
Commutative Ring ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Bilinear Form ◽  
Semisimple Lie Algebra ◽  
Direct Sum ◽  
Rings With Involution ◽  
Torsion Free

In a ringRwith involution whose symmetric elementsSare central, the skew-symmetric elementsKform a Lie algebra over the commutative ringS. The classification of such rings which are2-torsion free is equivalent to the classification of Lie algebrasKoverSequipped with a bilinear formfthat is symmetric, invariant and satisfies[[x,y],z]=f(y,z)x−f(z,x)y. IfSis a field of char≠2,f≠0anddimK>1thenKis a semisimple Lie algebra if and only iffis nondegenerate. Moreover, the derived algebraK′is either the pure quaternions overSor a direct sum of mutually orthogonal abelian Lie ideals ofdim≤2.

scholarly journals Invariant metrics on central extensions of quadratic Lie algebras

2019 ◽  
Vol 19 (12) ◽  
pp. 2050224
Author(s):  
R. García-Delgado ◽  
G. Salgado ◽  
O. A. Sánchez-Valenzuela
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Bilinear Form ◽  
General Structure ◽  
Invariant Metrics ◽  
Invariant Metric ◽  
Central Extensions ◽  
Direct Sum ◽  
Quadratic Lie Algebra

A quadratic Lie algebra is a Lie algebra endowed with a symmetric, invariant and nondegenerate bilinear form; such a bilinear form is called an invariant metric. The aim of this work is to describe the general structure of those central extensions of quadratic Lie algebras which in turn have invariant metrics. The structure is such that the central extensions can be described algebraically in terms of the original quadratic Lie algebra, and geometrically in terms of the direct sum decompositions that the invariant metrics involved give rise to.

Trace Forms on Lie Algebras

1962 ◽  
Vol 14 ◽  
pp. 553-564 ◽  
Author(s):  
Richard Block
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Bilinear Form ◽  
Symmetric Bilinear Form ◽  
Trace Form ◽  
Finite Degree ◽  
Trace Forms ◽  
Matrix Products ◽  
The Matrix ◽  
Image Position

If L is a Lie algebra with a representation Δ a→aΔ (a in L) (of finite degree), then by the trace form f = fΔ of Δ is meant the symmetric bilinear form on L obtained by taking the trace of the matrix products:Then f is invariant, that is, f is symmetric and f(ab, c) — f(a, bc) for all a, b, c in L. By the Δ-radical L⊥ = L⊥ of L is meant the set of a in L such that f(a, b) = 0 for all b in L. Then L⊥ is an ideal and f induces a bilinear form , called a quotient trace form, on L/L⊥. Thus an algebra has a quotient trace form if and only if there exists a Lie algebra L with a representation Δ such that

Non-degenerate Invariant Bilinear Forms on Lie Color Algebras

2010 ◽  
Vol 17 (03) ◽  
pp. 365-374 ◽  
Author(s):  
Song Wang ◽  
Linsheng Zhu
Keyword(s):  
Lie Algebras ◽  
Bilinear Form ◽  
Bilinear Forms ◽  
Sufficient Condition ◽  
Invariant Bilinear Form ◽  
Lie Color Algebras ◽  
Double Extension

In this paper, we study Lie color algebras 𝔤 with a non-degenerate color-symmetric, 𝔤-invariant bilinear form B, such a (𝔤,B) is called a quadratic Lie color algebra. Our first result generalizes the notion of double extensions to quadratic Lie color algebras. This notion was introduced by Medina and Revoy to study quadratic Lie algebras. In the second theorem, we give a sufficient condition for a quadratic Lie color algebra to be a quadratic Lie color algebra by double extension. At last, we generalize the notion of T*-extensions to Lie color algebras.

On Derivations of Lie Algebras

1976 ◽  
Vol 28 (1) ◽  
pp. 174-180 ◽  
Author(s):  
Stephen Berman
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Bilinear Form ◽  
Characteristic Zero ◽  
Killing Form ◽  
Simple Lie Algebras ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Finite Dimensional Case ◽  
Cartan Matrices

A well known result in the theory of Lie algebras, due to H. Zassenhaus, states that if is a finite dimensional Lie algebra over the field K such that the killing form of is non-degenerate, then the derivations of are all inner, [3, p. 74]. In particular, this applies to the finite dimensional split simple Lie algebras over fields of characteristic zero. In this paper we extend this result to a class of Lie algebras which generalize the split simple Lie algebras, and which are defined by Cartan matrices (for a definition see § 1). Because of the fact that the algebras we consider are usually infinite dimensional, the method we employ in our investigation is quite different from the standard one used in the finite dimensional case, and makes no reference to any associative bilinear form on the algebras.

Split Lie algebras of order 3

2019 ◽  
Vol 19 (04) ◽  
pp. 2050070
Author(s):  
Antonio J. Calderón ◽  
Rosa M. Navarro ◽  
José M. Sánchez
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Linear Subspace ◽  
Type Theorem ◽  
Natural Generalization ◽  
Lie Superalgebras ◽  
Direct Sum ◽  
The Family ◽  
Split Lie Algebra

We introduce the class of split Lie algebras of order 3 as the natural generalization of split Lie superalgebras and split Lie algebras. By means of connections of roots, we show that such a split Lie algebra of order 3 is of the form [Formula: see text] with [Formula: see text] a linear subspace of [Formula: see text] and any [Formula: see text] a well-described (split) ideal of [Formula: see text] satisfying [Formula: see text], with [Formula: see text], if [Formula: see text]. Additionally, under certain conditions, the (split) simplicity of the algebra is characterized in terms of the connections of nonzero roots, and a second Wedderburn type theorem for the class of split Lie algebras of order 3 (asserting that [Formula: see text] is the direct sum of the family of its (split) simple ideals) is stated.

DERIVATION ALGEBRAS OF RINGS RELATED TO HEISENBERG ALGEBRAS

2004 ◽  
Vol 03 (02) ◽  
pp. 181-191 ◽  
Author(s):  
JEFFREY BERGEN
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Quotient Ring ◽  
Finite Codimension ◽  
Enveloping Algebras ◽  
Cartan Type ◽  
Direct Sum ◽  
Heisenberg Algebras ◽  
Martindale Quotient Ring ◽  
The Ideal

In this paper, we will determine the Lie algebra of derivations of rings which are generalizations of the enveloping algebras of Heisenberg Lie algebras. First, we will determine which derivations are X-inner and also determine which elements in the Martindale quotient ring induce X-inner derivations. Then, we will show that the Lie algebra of derivations is the direct sum of the ideal of X-inner derivations and a subalgebra which is isomorphic to a subalgebra of finite codimension in a Cartan type Lie algebra.

The q-Analog Klein Bottle Lie Algebra

2014 ◽  
Vol 21 (04) ◽  
pp. 561-574
Author(s):  
Cuipo Jiang ◽  
Jingjing Jiang ◽  
Yufeng Pei
Keyword(s):  
Lie Algebra ◽  
Bilinear Form ◽  
Central Extension ◽  
Klein Bottle ◽  
Universal Central Extension ◽  
Finitely Generated ◽  
Invariant Bilinear Form ◽  
Infinite Dimensional ◽  
Simple Lie Algebra ◽  
Infinite Dimensional Lie Algebra

In this paper, we study an infinite-dimensional Lie algebra ℬq, called the q-analog Klein bottle Lie algebra. We show that ℬq is a finitely generated simple Lie algebra with a unique (up to scalars) symmetric invariant bilinear form. The derivation algebra and the universal central extension of ℬq are also determined.

VI.—Complex Four-dimensional Lie Algebras

1958 ◽  
Vol 65 (1) ◽  
pp. 72-83
Author(s):  
E. W. Wallace
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Canonical Forms ◽  
Complete Set

SynopsisCanonical forms of the four-dimensional complex Lie algebras are obtained by considering the roots of certain well-defined vectors of the algebras. A complete set of characters of the algebras is also given, enabling any given four-dimensional complex Lie algebra to be identified with one of the canonical forms.

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