Related papers: On a quarternification of complex Lie algebras
Starting from the four normed division algebras - the real numbers, complex numbers, quaternions and octonions - a systematic procedure gives a 3-cocycle on the Poincare Lie superalgebra in dimensions 3, 4, 6 and 10. A related procedure…
We realize the enveloping algebra of the positive part of a semisimple complex Lie algebra as a convolution algebra of constructible functions on module varieties of some Iwanaga-Gorenstein algebras of dimension 1.
Over an algebraically closed fields, an alternative to the method due to Kostrikin and Shafarevich was recently suggested. It produces all known simple finite dimensional Lie algebras in characteristic p>2. For p=2, we investigate one of…
In this paper we describe all Lie bialgebra structures on the polynomial Lie algebra $\mathbf{g}[u]$, where $\mathbf{g}$ is a simple, finite dimensional, complex Lie algebra. The results are based on an unpublished paper Montaner and…
Relying on the classification of simple Lie algebras over algebraically closed fields of characteristic $>3$, we show that any finite-dimensional central simple 5-graded Lie algebra over a field $k$ of characteristic $\neq 2,3$ is a simple…
We give a uniform interpretation of the classical continuous Chebyshev's and Hahn's orthogonal polynomials of discrete variable in terms of Feigin's Lie algebra gl(N), where N is any complex number. One can similarly interpret Chebyshev's…
The set HLie(n) of the n-dimensional Hom-Lie algebras over an algebraically closed field of characteristic zero is provided with a structure of algebraic subvariety of the affine plane of dimension n^2(n-1)/2}. For n=3, these two sets…
The most general possible central extensions of two whole families of Lie algebras, which can be obtained by contracting the special pseudo-unitary algebras su(p,q) of the Cartan series A_l and the pseudo-unitary algebras u(p,q), are…
Let $G$ be a complex reductive algebraic group, $g$ its Lie algebra and $h$ a reductive subalgebra of $g$, $n$ a positive integer. Consider the diagonal actions $G:g^n, N_G(h):h^n$. We study a relation between the algebra $C[h^n]^{N_G(h)}$…
We illustrate some simple ideas that can be used for obtaining a classification of small-dimensional solvable Lie algebras.Using these we obtain the classification of 3 and 4 dimensional solvable Lie algebras (over fields of any…
If $L$ is a semisimple Lie algebra of vector fields on R^N with a split Cartan subalgebra C, then it is proved that the dimension of the generic orbit of C coincides with the dimension of C. As a consequence one obtains a local canonical…
Recently, it has been shown how to perform the quantum hamiltonian reduction in the case of general $sl(2)$ embeddings into Lie (super)algebras, and in the case of general $osp(1|2)$ embeddings into Lie superalgebras. In another development…
We call a finite-dimensional complex Lie algebra $\mathfrak{g}$ strongly rigid if its universal enveloping algebra $\Ug$ is rigid as an associative algebra, i.e. every formal associative deformation is equivalent to the trivial deformation.…
We generalize several important results from the perturbation theory of linear operators to the setting of semisimple orthogonal symmetric Lie algebras. These Lie algebras provide a unifying framework for various notions of matrix…
The powerful group theoretical formalism of potential algebras is extended to non-Hermitian Hamiltonians with real eigenvalues by complexifying so(2,1), thereby getting the complex algebra sl(2,\C) or $A_1$. This leads to new types of both…
The Drinfeld double structure underlying the Cartan series An, Bn, Cn, Dn of simple Lie algebras is discussed. This structure is determined by two disjoint solvable subalgebras matched by a pairing. For the two nilpotent positive and…
Let $\mathfrak{g}$ be a vector space and $[,],[,]'$ be a pair of Lie brackets on $\mathfrak{g}$. By definition they are compatible if $[,]+[,]'$ is again a Lie bracket. Such pairs play important role in bihamiltonian and $r$-matrix…
We determine the Lie superalgebras that are graded by the root systems of the basic classical simple Lie superalgebras of type A$(m,n)$.
Leibniz algebras are certain generalization of Lie algebras. In this paper we give classification of non-Lie solvable (left) Leibniz algebras of dimension $\leq 8$ with one dimensional derived subalgebra. We use the canonical forms for the…
Given a symmetric non degenerated bilinear form b on a vector space V, G. Pinczon and R. Ushirobira defined a bracket {,} on the space of multilinear skewsymmetric forms on V. With this bracket, the quadratic Lie algebra structure equation…