Any Lie algebra equipped with a symplectic form can be equipped with an affine structure. On the other hand there exist (2p + 1)-dimensional Lie algebras with contact form and no affine structure. But each nilpotent contact Lie algebra is a one-dimensional central extension of a symplectic algebra. The aim of this work is to study how we can extend, under certain conditions, the symplectic stuc...

In her thesis [RB], Ranee Brylinski (then Gupta) studied the orbit structure of the projective variety of abelian subalgebras of a …xed dimension, k, in a simple Lie algebra, g, over C under its adjoint group, G. Fix a Borel subalgebra, b, of g and let B be the closed subgroup of G corresponding to b. Then the Borel …xed point theorem implies that the closed G-orbits are precisely the orbits of...

The classical Tits construction provides models of the exceptional simple Lie algebras in terms of a unital composition algebra and a degree three simple Jordan algebra. A couple of actions of the symmetric group S4 on this construction are given. By means of these actions, the models provided by the Tits construction are related to models of the exceptional Lie algebras obtained from two diffe...

The bosonic sector of various supergravity theories reduces to a homogeneous space G/H in three dimensions. The corresponding algebras g are simple for (half-)maximal supergravity, but can be semi-simple for other theories. We extend the existing literature on the Kac–Moody extensions of simple Lie algebras to the semi-simple case. Furthermore, we argue that for N = 2 supergravity the simple al...

Abstract. In this paper we classify a linear family of Lie brackets on the space of rectangular matrices Mat(n×m, K) and we give an analogue of the Ado’s Theorem. We give also a similar classification on the algebra of the square matrices Mat(n, K) and as a consequence, we prove that we can’t built a faithful representation of the (2n + 1)-dimensional Heisenberg Lie algebra Hn in a vector space...

This essay is intended to be a self-contained if rather brief introduction to Lie algebras. The first several sections deal with Lie algebras as spaces of invariant vector fields on a Lie group. The very first sections are concerned with different ways to consider vector fields. This early treatment is intended, among other things, to motivate the eventual introduction of abstract Lie algebras,...

Abstract. In this paper we classify a linear family of Lie brackets on the space of rectangular matrices Mat(n×m, K) and we give an analogue of the Ado’s Theorem. We give also a similar classification on the algebra of the square matrices Mat(n, K) and as a consequence, we prove that we can’t built a faithful representation of the (2n + 1)-dimensional Heisenberg Lie algebra Hn in a vector space...

Groups of individuals can sometimes make more accurate judgments than the average individual could make alone. We tested whether this group advantage extends to lie detection, an exceptionally challenging judgment with accuracy rates rarely exceeding chance. In four experiments, we find that groups are consistently more accurate than individuals in distinguishing truths from lies, an effect tha...

6 Lie Derivatives and the Commutator Revisited 21 6.1 Integral Curves . . . . . . . . . . . . . . . . . . . . . . . . . . 21 6.2 Congruence of Curves . . . . . . . . . . . . . . . . . . . . . . . 21 6.3 The Commutator Revisited: A Geometric Interpretation . . . 22 6.4 Lie Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . 24 6.5 Lie Derivatives of a Function . . . . . . . . . . . ....