Related papers: Assembling Lie Algebras from Lieons

Leibniz algebras are certain generalization of Lie algebras. In this paper we survey the important results in Leibniz algebras which are analogs of corresponding results in Lie algebras. In particular we highlight the differences between…

We use the Lie coalgebra and configuration pairing framework presented previously by Sinha and Walter to derive a new, left-normed monomial basis for free Lie algebras (built from associative Lyndon-Shirshov words), as well as a dual…

We try to write (generic) elements of classical simple Lie algebras as sums of as few elements of the orbit C of long root vectors as possible. If the Lie algebra in question is sl_n or sp_n, then C consists of all rank 1 matrices in the…

Let g be a finite-dimensional complex semi simple Lie algebra. We present a new calculation of the continuous cohomology of the Lie algebra z g[[z]]. In particular, we shall give an explicit formula for the Laplacian on the Lie algebra…

Realizations of four dimensional Lie algebras as vector fields in the plane are explicitly constructed. Fourth order ordinary differential equations which admit such Lie symmetry algebras are derived. The route to their integration is…

We study post-Lie algebra structures on pairs of Lie algebras (g,n), and prove existence results for the case that one of the Lie algebras is semisimple. For semisimple g and solvable n we show that there exist no post-Lie algebra…

In this paper, we shall give a way to construct a graded Lie algebra $L(\mathfrak{g},\rho,V,{\cal V},B_0)$ from a standard pentad $(\mathfrak{g},\rho,V,{\cal V},B_0)$ which consists of a Lie algebra $\mathfrak{g}$ which has a non-degenerate…

Uniform Lie algebras are combinatorially defined two-step nilpotent Lie algebras which can be used to define Einstein solvmanifolds. These Einstein spaces often have nontrivial isotropy groups. We derive basic properties of uniform Lie…

Let $\mathfrak{g}$ be a reductive Lie algebra over an algebraically closed, characteristic zero field or over $\mathbb{R}$. Let $\mathfrak{q}$ be a parabolic subalgebra of $\mathfrak{g}$. We characterize the derivations of $\mathfrak{q}$ by…

We present structural properties of Lie algebras admitting symmetric, invariant and nondegenerate bilinear forms. We show that these properties are not satisfied by nilradicals of parabolic subalgebras of real split forms of complex simple…

We characterise Lie groups with bi-invariant bargmannian, galilean or carrollian structures. Localising at the identity, we show that Lie algebras with ad-invariant bargmannian, carrollian or galilean structures are actually determined by…

Lie quasi-bialgebras are natural generalisations of Lie bialgebras introduced by Drinfeld. To any Lie quasi-bialgebra structure of finite-dimensional (G, \mu, \gamma ,\phi ?), correspond one Lie algebra structure on D = G\oplus G*, called…

We define a solvable extension of the graph 2-step nilpotent Lie algebras of [5] by adding elements corresponding to the 3-cliques of the graph. We study some of their basic properties and we prove that two such Lie algebras are isomorphic…

We introduce the class of split Lie-Rinehart algebras as the natural extension of the one of split Lie algebras. We show that if $L$ is a tight split Lie-Rinehart algebra over an associative and commutative algebra $A,$ then $L$ and $A$…

A Lie 2-algebra is a linear category equipped with a functorial bilinear operation satisfying skew-symmetry and Jacobi identity up to natural transformations which themselves obey coherence laws of their own. Functors and natural…

Let i be a homomorphism of the multiplicative group into a connected reductive algebraic group over C. Let G^i be the centralizer of the image i. Let LG be the Lie algebra of G and let L_nG (n integer) be the summands in the direct sum…

Admissible structure constants related to the dual Lie superalgebras of particular Lie superalgebra $({\cal C}^3 + {\cal A})$ are found by straightforward calculations from the matrix form of super Jacobi and mixed super Jacobi identities…

The expansion of a Lie algebra entails finding a new, bigger algebra G, through a series of well-defined steps, from an original Lie algebra g. One incarnation of the method, the so-called S-expansion, involves the use of a finite abelian…

We study the adjoint cohomology of perfect Lie algebras over the complex numbers. For the family of perfect Lie algebras $\mathfrak{g}=\mathfrak{sl}_2(\Bbb C)\ltimes V_m$ we obtain some explicit results for $H^k(\mathfrak{g},\mathfrak{g})$…

In this paper we give a complete classification of two-step nilpotent Leibniz algebras in terms of Kronecker modules associated with pairs of bilinear forms. In particular, we describe the complex and the real case of the indecomposable…