Morphisms deformations and abelian extensions of differential Lie algebras

In this article we extend and adapt several results on extensions of Lie algebras to topological Lie algebras over topological fields of characteristic zero. In particular, we describe the set of equivalence classes of extensions of the Lie algebra 𝔤 by the Lie algebra 𝔫 as a disjoint union of affine spaces with translation group H 2(𝔤, 𝔷(𝔫))[S], where [S] denotes the equivalence class of the continuous outer action S : 𝔤 → der sp;𝔫. We also discuss topological crossed modules and explain how they are related to extensions of Lie algebras by showing that any continuous outer action gives rise to a crossed module whose obstruction class in H 3(𝔤, 𝔷(𝔫)) S is the characteristic class of the corresponding crossed module. The correspondence between crossed modules and extensions further leads to a description of 𝔫-extensions of 𝔤 in terms of certain 𝔷(𝔫)-extensions of a Lie algebra which is an extension of 𝔤 by 𝔫/𝔷(𝔫). We discuss several types of examples, describe applications to Lie algebras of vector fields on principal bundles, and in two appendices we describe the set of automorphisms and derivations of topological Lie algebra extensions.

In this work, we consider extensions of solvable Lie algebras with naturally graded filiform nilradicals. Note that there exist two naturally graded filiform Lie algebras [Formula: see text] and [Formula: see text] We find all one-dimensional extensions of solvable Lie algebras with nilradical [Formula: see text]. We prove that there exists a unique non-split central extension of solvable Lie algebras with nilradical [Formula: see text] of maximal codimension. Moreover, all one-dimensional extensions of solvable Lie algebras with nilradical [Formula: see text] whose codimension is equal to one are found and we compared these solvable algebras with the solvable algebras with nilradicals that are one-dimensional central extension of algebra [Formula: see text].

In this paper we give some results on the extensions of Lie algebras, with emphasis on the case of prime characteristic, although part of the paper is also of interest at characteristic 0. An extension of a Lie algebra L is a pair (E, π), where £ is a Lie algebra and π is a homomorphism of E onto L. The kernel K of the extension is ker π.

The purpose of this paper is to show how central extensions of (possibly infinite-dimensional) Lie algebras integrate to central extensions of étale Lie 2-groups in the sense of [Get09, Hen08]. In finite dimensions, central extensions of Lie algebras integrate to central extensions of Lie groups, a fact which is due to the vanishing of \pi_2 for each finite-dimensional Lie group. This fact was used by Cartan (in a slightly other guise) to construct the simply connected Lie group associated to each finite-dimensional Lie algebra. In infinite dimensions, there is an obstruction for a central extension of Lie algebras to integrate to a central extension of Lie groups. This obstruction comes from non-trivial \pi_2 2 for general Lie groups. We show that this obstruction may be overcome by integrating central extensions of Lie algebras not to Lie groups but to central extensions of étale Lie 2-groups. As an application, we obtain a generalization of Lie’s Third Theorem to infinite-dimensional Lie algebras.

Non-abelian cohomology of extensions of Lie algebras as Deligne groupoid

An indecomposable Lie group with Riemannian bi-invariant metric is always simple and hence Einstein. For indefinite metrics this is no longer true, not even for simple Lie groups. We study the question of whether a semi-Riemannian bi-invariant metric is conformal to an Einstein metric. We obtain results for all three cases in the structure theorem by Medina and Revoy for indecomposable metric Lie algebras: the case of simple Lie algebras, and the cases of double extensions of metric Lie algebras by $\mathbb{R}$ or a simple Lie algebra. Simple Lie algebras are conformally Einstein precisely when they are Einstein, or when equal to $\mathfrak{sl}_2\mathbb{C}$ and conformally flat. Double extensions of metric Lie algebras by simple Lie algebras of rank greater than one are never conformally Einstein, and neither are double extensions of Lorentzian oscillator algebras, whereas the oscillator algebras themselves are conformally Einstein. Our results give a complete answer to the question of which metric Lie algebras in Lorentzian signature and in signature (2,n-2) are conformally Einstein.

5-dimensional indecomposable contact Lie algebras as double extensions

Symplectic structures on quadratic Lie algebras

Deformations, cohomologies and integrations of relative difference Lie algebras

In this paper, we give Maurer-Cartan characterizations as well as a cohomology theory for compatible Lie algebras. Explicitly, we first introduce the notion of a bidifferential graded Lie algebra and thus give Maurer-Cartan characterizations of compatible Lie algebras. Then we introduce a cohomology theory of compatible Lie algebras and use it to classify infinitesimal deformations and abelian extensions of compatible Lie algebras. In particular, we introduce the reduced cohomology of a compatible Lie algebra and establish the relation between the reduced cohomology of a compatible Lie algebra and the cohomology of the corresponding compatible linear Poisson structures introduced by Dubrovin and Zhang in their study of bi-Hamiltonian structures. Finally, we use the Maurer-Cartan approach to classify nonabelian extensions of compatible Lie algebras.

Non-abelian extensions of Rota-Baxter Lie algebras and inducibility of automorphisms

We introduce the extended Freudenthal–Rosenfeld–Tits magic square based on six algebras: the reals , complexes , ternions , quaternions , sextonions and octonions . The sextonionic row/column of the magic square appeared previously and was shown to yield the non-reductive Lie algebras, , , , and , for and respectively. The fractional ranks are used to denote the semi-direct extension of the simple Lie algebra in question by a unique (up to equivalence) Heisenberg algebra. The ternionic row/column yields the non-reductive Lie algebras, , , , and , for and respectively. The fractional ranks here are used to denote the semi-direct extension of the semi-simple Lie algebra in question by a unique (up to equivalence) nilpotent ‘Jordan’ algebra. We present all possible real forms of the extended magic square. It is demonstrated that the algebras of the extended magic square appear quite naturally as the symmetries of supergravity Lagrangians. The sextonionic row (for appropriate choices of real forms) gives the non-compact global symmetries of the Lagrangian for the maximal , magic and magic non-supersymmetric theories, obtained by dimensionally reducing the parent theories on a circle, with the graviphoton left undualised. In particular, the extremal intermediate non-reductive Lie algebra (which is not a subalgebra of ) is the non-compact global symmetry algebra of , supergravity as obtained by dimensionally reducing , supergravity with symmetry on a circle. On the other hand, the ternionic row (for appropriate choices of real forms) gives the non-compact global symmetries of the Lagrangian for the maximal , magic and magic non-supersymmetric theories, as obtained by dimensionally reducing the parent theories on a circle. In particular, the Kantor–Koecher–Tits intermediate non-reductive Lie algebra is the non-compact global symmetry algebra of , supergravity as obtained by dimensionally reducing , supergravity with symmetry on a circle.

We study the problem of quadruple extensions of simple Lie algebras. We find that, adding a new simple root α+4, it is not possible to have an extended Kac-Moody algebra described by a Dynkin-Kac diagram with simple links and no loops between the dots, while it is possible if α+4 is a Borcherds imaginary simple root. We also comment on the root lattices of these new algebras. The folding procedure is applied to the simply laced triple extended Lie algebras, obtaining all the nonsimply laced ones. Nonstandard extension procedures for a class of Lie algebras are proposed. It is shown that the two-extensions of E8, with a dot simply linked to the Dynkin-Kac diagram of E9, are rank 10 subalgebras of E10. Finally the simple root systems of a set of rank 11 subalgebras of E11, containing as sub-algebra E10, are explicitly written.

Journal Article Twisted modules for vertex operator algebras and Bernoulli polynomials Get access B. Doyon, B. Doyon Search for other works by this author on: Oxford Academic Google Scholar J. Lepowsky, J. Lepowsky Search for other works by this author on: Oxford Academic Google Scholar A. Milas A. Milas Search for other works by this author on: Oxford Academic Google Scholar International Mathematics Research Notices, Volume 2003, Issue 44, 2003, Pages 2391–2408, https://doi.org/10.1155/S1073792803130863 Published: 01 January 2003 Article history Published: 01 January 2003 Received: 17 March 2003 Accepted: 27 July 2003

We describe explicitly in terms of generators and relations the universal central extension of infinite-dimensional superelliptic affine Lie algebras with a finite-dimensional simple Lie algebra and the coordinate ring where and p(t) is a polynomial with distinct roots.