Related papers: Structures Preserved by Exceptional Lie Algebras

We construct a universal continuous invariant bilinear form for the Lie algebra of compactly supported sections of a Lie algebra bundle in a topological sense. Moreover we construct a universal continuous central extension of a current…

We show there is a class of symplectic Lie algebra representations over any field of characteristic not 2 or 3 that have many of the exceptional algebraic and geometric properties of both symmetric three forms in two dimensions and…

We observe that, up to conjugation, a majority of symmetric higher order ODEs (ordinary differential equations) and ODE systems have only fiber-preserving point symmetries. By exploiting Lie's classification of Lie algebras of vector…

Given a finite-dimensional, complex simple Lie algebra we exhibit an integral form for the universal enveloping algebra of its map algebra, and an explicit integral basis for this integral form. We also produce explicit commutation formulas…

We study the cohomology of Lie superalgebras for the full complex of forms: superforms, pseudoforms and integral forms. We use the technique of spectral sequences to abstractly compute the Chevalley-Eilenberg cohomology. We first focus on…

Let L be a restricted Lie algebra over a field of positive characteristic. We survey the known results about the Lie structure of the restricted enveloping algebra u(L) of L. Related results about the structure of the group of units and the…

In this paper, we classify solvable Lie algebras of dimensions $\leq 8$ endowed with a nondegenerate invariant symmetric bilinear form over an algebraically closed field. This classification (up to isometrically isomorphisms) is mainly…

The irreducible tensor bases of exceptional Lie algebras G2, F4 and E6 are built by grouping their Cartan-Weyl bases according to the respective chains G2> SO(3) * SO(3), F4 > SO(3)*SO(3)*SO(3)*SO(3) and E6> SO(3)*SO(3)*SO(3)*SO(3). The…

We generate by computer a basis of invariants for the fundamental representations of the exceptional Lie groups E(6) and E(7), up to degree 18. We discuss the relevance of this calculation for the study of supersymmetric gauge theories, and…

Let L be a Lie superalgebra over a fled of characteristic p\neq 2 with enveloping algebra U(L) or let L be a restricted Lie superalgebra over a field of characteristic p > 2 with restricted enveloping algebra u(L). In this note, we…

Some fine gradings on the exceptional Lie algebras $\mathfrak{e}_6$, $\mathfrak{e}_7$ and $\mathfrak{e}_8$ are described. This list tries to be exhaustive.

The co-Lie structures compatible with the osp(2|2) Lie super algebra structure are investigated and found to be all of coboundary type. The corresponding classical r-matrices are classified into several disjoint families. The osp(1|2)+u(1)…

In this work we state a version of the double extension for homogeneous quadratic Lie super algebras that includes even and odd cases. We prove that any indecomposable, non-simple and homogeneous quadratic Lie super algebra is obtained by…

The normal form for a system of ode's is constructed from its polynomial symmetries of the linear part of the system, which is assumed to be semi-simple. The symmetries are shown to have a simple structure such as invariant function times…

For the $p$-dimensional filiform Lie algebra ${\mathfrak m}_2(p)$ over a field ${\mathbb F}$ of prime characteristic $p\ge 5$ with nonzero Lie brackets $[e_1,e_i] = e_{i+1}$ for $1<i<p$ and $[e_2,e_i]=e_{i+2}$ for $2<i<p-1$, we show that…

The space K^n of all n-dimensional { Lie} algebras has a natural non-Hausdorff topology k^n, which has characteristic limits, called transitions, A -> B, between distinct Lie algebras A and B. The entity of these transitions are the natural…

Exceptional field theory (EFT) gives a geometric underpinning of the U-duality symmetries of M-theory. In this talk I give an overview of the surprisingly rich algebraic structures which naturally appear in the context of EFT. This includes…

Let $A=F[x,y]$ be the polynomial algebra on two variables $x,y$ over an algebraically closed field $F$ of characteristic zero. Under the Poisson bracket, $A$ is equipped with a natural Lie algebra structure. It is proven that the maximal…

In this paper we consider Lie superalgebras decomposable as the sum of two proper subalgebras. Any of these algebras has the form of the vector space sum $L=A+B$ where $A$ and $B$ are proper simple subalgebras which need not be ideals of…

The paper studies the structure of restricted Leibniz algebras. More specifically speaking, we first give the equivalent definition of restricted Leibniz algebras, which is by far more tractable than that of a restricted Leibniz algebras in…