Let $L$ be a finite-dimensional Lie algebra. We say a subalgebra $H$ of $L$ is permutably complemented in $L$ if there is a subalgebra $K$ of $L$ such that $L=H+K$ and $Hcap K=0$. Also, if every subalgebra of $L$ is permutably complemented in $L$, then $L$ is called completely factorisable. In this article, we consider the influence of these concepts on the structure of a Lie algebra, in partic...

A reductive Lie algebra g is one that can be written C(g) ⊕ [g,g], where C(g) denotes the center of g. Equivalently, for any ideal a, there is another ideal b such that g = a⊕ b. A Cartan subalgebra of g is a subalgebra h that is maximal with respect to being abelian and having ad X being semisimple for all X ∈ h. For a reductive group, h = C(g) ⊕ h′, where h′ is a Cartan subalgebra of the semi...

The group gradings on the symmetric composition algebras over arbitrary fields are classified. Applications of this result to gradings on the exceptional simple Lie algebras are considered too.

Let g be a complex semisimple Lie algebra, with a Bore1 subalgebra b c g and Cartan subalgebra h c b. In classifying the finite dimensional representations of g, Cartan showed that any simple finite dimensional g-module has a generating element u, annihilated by n = [b, b], on which h acts by a linear form I E h*. Such an element is called a primitive vector (for the module). Harish-Chandra [9]...

Lying is typically more cognitively demanding than truth telling. Yet, recent cognitive models of lying propose that lying can be just as easy as truth telling, depending on contextual factors. In line with this idea, research has shown that the cognitive cost of deception decreases when people frequently respond deceptively, while it increases when people rarely respond deceptively (i.e., the ...

In this contributed presentation, we discuss and compare the mutually opposite procedures of deformations and contractions of Lie algebras. We suggest that with appropriate combinations of both procedures one may construct new Lie algebras. We first discuss low-dimensional Lie algebras and illustrate thereby that whereas for every contraction there exists a reverse deformation, the converse is ...

This paper encloses a complete and explicit description of the derivations of the Lie algebra D(M) of all linear differential operators of a smooth manifold M , of its Lie subalgebra D 1 (M) of all linear first-order differential operators of M , and of the Poisson algebra S(M) = Pol(T * M) of all polynomial functions on T * M, the symbols of the operators in D(M). It turns out that, in terms o...

We present an explicit description of the 'fine group gradings' (i.e. group gradings which cannot be further refined) of the real forms of the semisimple Lie algebras sl(4, C), sp(4, C), and o(4, C). All together 12 real Lie algebras are considered, and the total of 44 of their fine group gradings are listed. The inclusions sl(4, C) ⊃ sp(4, C) ⊃ o(4, C) are an important tool in our presentation...

This article critically reviews techniques and theories relevant to the emerging field of "lie detection by inducing cognitive load selectively on liars." To help these techniques benefit from past mistakes, we start with a summary of the polygraph-based Controlled Question Technique (CQT) and the major criticisms of it made by the National Research Council (2003), including that it not based o...

The algebraic cohoroology and the spectral sequences for a Z2-graded Lie algebra are briefly reviewed. The reducibility property of a strongly semisiiaple Lie superalgebra is established. The role of second and third cohomologies in the deformation of a Lie superalgebra is discussed. Using spectral sequences, the second cohomology of full BRS algebra is shown to be W and the third cohomology be...