Quadratic Lie Algebras Research Articles

On quadratic Hom-Lie algebras with twist maps in their centroids and their relationship with quadratic Lie algebras

Hom-Lie algebras having non-invertible twist maps in their centroids are studied. Central extensions of Hom-Lie algebras having these properties are obtained and shown how the same properties are preserved. Conditions are given so that the produced central extension has an invariant metric with respect to its Hom-Lie product making its twist map self-adjoint when the original Hom-Lie algebra has such a metric. This work is focused on algebras with these properties and following Benayadi and Makhlouf we call them quadratic Hom-Lie algebras. It is shown how a quadratic Hom-Lie algebra gives rise to a quadratic Lie algebra and that the Lie algebra associated to the given Hom-Lie central extension is a Lie algebra central extension of it. It is also shown that if the Hom-Lie product is not a Lie product, there exists a non-abelian algebra, which is in general non-associative too, the commutator of whose product is precisely the Hom-Lie product of the Hom-Lie central extension. Moreover, the algebra whose commutator realizes this Hom-Lie product is shown to be simple if the associated Lie algebra is nilpotent. Non-trivial examples are provided.

Complete description of invariant, associative pseudo-Euclidean metrics on left Leibniz algebras via quadratic Lie algebras

A pseudo-Euclidean non-associative algebra (g,•) is a finite dimensional algebra over a field K that has a metric, i.e., a bilinear, symmetric, and non-degenerate form 〈,〉. The metric is considered L-invariant (resp. R-invariant) if all left multiplications (resp. right multiplications) are skew-symmetric. The metric is called associative if 〈u•v,w〉=〈u,v•w〉 for all u,v,w∈g. These three notions coincide when g is a Lie algebra and in this case g endowed with the metric is known as a quadratic Lie algebra.This paper provides a complete description of L-invariant, R-invariant, or associative pseudo-Euclidean metrics on left Leibniz algebras over a commutative field of characteristic zero. It shows that a left Leibniz algebra with an associative metric is also right Leibniz and can be obtained easily from its underlying Lie algebra, which is a quadratic Lie algebra. Additionally, it shows that at the core of a left Leibniz algebra endowed with a L-invariant or R-invariant metric, there are two Lie algebras with one quadratic and the left Leibniz algebra can be built from these Lie algebras. We derive many important results from this complete description. Finally, the paper provides a list of left Leibniz algebras with an associative metric up to dimension 6, as well as a list of left Leibniz algebras with an L-invariant metric, up to dimension 4, and R-invariant metric up to dimension 5.

On pseudo-Hermitian quadratic nilpotent lie algebras

We study nilpotent Lie algebras endowed with a complex structure and a quadratic structure which is pseudo-Hermitian for the given complex structure. We propose several methods to construct such Lie algebras and describe a method of double extension by planes to get an inductive description of all of them. As an application, we give a complete classification of nilpotent quadratic Lie algebras where the metric is Lorentz-Hermitian and we fully classify all nilpotent pseudo-Hermitian quadratic Lie algebras up to dimension 8 and their inequivalent pseudo-Hermitian metrics.

Quadratic Lie algebras with 2-plectic structures

We study the existence of 2-plectic structures on Lie algebras which admit an ad-invariant non-degenerate symmetric bilinear form, frequently called quadratic Lie algebras. It is well-known that every centerless quadratic Lie algebra admits a 2-plectic form but not many quadratic examples with nontrivial center are known. In this paper we give several constructions to obtain large families of 2-plectic quadratic Lie algebras with nontrivial center, many of them among the class of nilpotent Lie algebras. We give some sufficient conditions to assure that certain extensions of 2-plectic quadratic Lie algebras result to be 2-plectic as well. We prove that every quadratic and symplectic Lie algebra with dimension greater than 4 also admits a 2-plectic form. Further, conditions to assure that one may find a 2-plectic which is exact on certain quadratic Lie algebras are also obtained.

Invariant metrics on current Lie algebras

In this work we state conditions for a current Lie algebra to admit an invariant metric, where is a quadratic Lie algebra and is an associative and commutative algebra with unit. We show that if is an indecomposable quadratic Lie algebra, then admits an invariant metric if and only if also admits an invariant metric. In addition, we prove a theorem similar to the double extension for , where is an indecomposable, nilpotent and quadratic Lie algebra.

Lagrangian and orthogonal splittings, quasitriangular Lie bialgebras, and almost complex product structures

We study complex product structures on quadratic vector spaces and on quadratic Lie algebras analyzing the Lagrangian and orthogonal splittings associated with them. We show that a Manin triple equipped with generalized metric G+B such that B is an O-operator with extension G of mass −1 can be turned into another Manin triple that admits also an orthogonal splitting in Lie ideals. Conversely, a quadratic Lie algebra orthogonal direct sum of a pair of anti-isomorphic Lie algebras, following similar steps as in the previous case, can be turned into a Manin triple admitting an orthogonal splitting into Lie ideals.

L∞-formality check for the Hochschild complex of certain universal enveloping algebras

We study the L∞-formality problem for the Hochschild complex of the universal enveloping algebra of some examples of Lie algebras such as Cartan-3-regular quadratic Lie algebras (for example semisimple Lie algebras and in more detail so(3)), and free Lie algebras. We show that for these examples formality in Kontsevich's sense does NOT hold, but we compute the L∞ structure on the cohomology given by homotopy transfer in certain cases.

(0,2) Mirror Symmetry on Homogeneous Hopf Surfaces

Abstract In this work, we find the first examples of (0,2) mirror symmetry on compact non-Kähler complex manifolds. For this, we follow Borisov’s approach to mirror symmetry using vertex algebras and the chiral de Rham complex. Our examples of (0,2) mirrors are given by pairs of Hopf surfaces endowed with a Bismut-flat pluriclosed metric. Requiring that the geometry is homogeneous, we reduce the problem to the study of Killing spinors on a quadratic Lie algebra and the construction of embeddings of the $N=2$ superconformal vertex algebra in the superaffine vertex algebra, combined with topological T-duality.

On pseudo-Euclidean Lie algebras whose Levi-Civita product is left Leibniz

We study a class of Lie algebras which contains the class of quadratic Lie algebras and the class of Milnor Lie algebras, namely, Lie algebras endowed a pseudo-Euclidean metric such its Levi-Civita product is left Leibniz. We call them Levi-Civita left Leibniz Lie algebras LCLL for short. We show that a Lie group (G,h) endowed with a left invariant pseudo-Riemannian such that the corresponding Lie algebra is LCLL is complete and locally symmetric. Moreover, there is a pointed Lie rack structure on G whose corresponding left Leibniz algebra is given by the Levi-Civita product. Euclidean LCLL Lie algebras are the product of a quadratic Lie algebra and a flat Euclidean Lie algebra. We develop an adapted version of the process of double extension to construct LCLL Lie algebras. We show that Lorentzian or flat LCLL Lie algebras can be obtained by this process.

Equivalent constructions of nilpotent quadratic Lie algebras

The double extension and the T⁎-extension are classical methods for constructing finite dimensional quadratic Lie algebras. The first one gives an inductive classification in characteristic zero, while the latest produces quadratic non-associative algebras (not only Lie) out of arbitrary ones in characteristic different from 2. The classification of quadratic nilpotent Lie algebras can also be reduced to the study of free nilpotent Lie algebras and their invariant forms. In this work we will establish an equivalent characterization among these three construction methods. This equivalence reduces the classification of quadratic 2-step nilpotent to that of trivectors in a natural way. In addition, theoretical results will provide simple rules for switching among them.

SYMPLECTIC DIRAC OPERATORS FOR LIE ALGEBRAS AND GRADED HECKE ALGEBRAS

The aim of this paper is to define a pair of symplectic Dirac operators (D+, D–) in an algebraic setting motivated by the analogy with the algebraic orthogonal Dirac operators in representation theory. We work in the settings of ℤ/2-graded quadratic Lie algebras 𝔤 = 𝔨 + 𝔭 and of graded affine Hecke algebras ℍ. In these contexts, we show analogues of the Parthasarathy’s formula for [D+, D–] and certain generalisations of the Casimir inequality.

Flat left-invariant pseudo-Riemannian metrics on quadratic Lie groups

A flat quadratic Lie algebra (g,〈,〉,k) is a Lie algebra g endowed with a flat pseudo-Euclidean metric 〈,〉 and a quadratic structure k. In geometrical terms, it is a Lie algebra of a Lie group endowed with both a flat left-invariant pseudo-Riemannian metric and a bi-invariant metric. It is known that if a Lie algebra admits a symplectic form and a quadratic structure then it is a flat quadratic Lie algebra. In this paper, we show that there exist non-abelian Lie algebras which admit a flat Lorentzian metric and a quadratic structure. If g is such Lie algebra, then we show that g is a trivial central extension of R⋊H2k+1 and [g,g]≃H2k+1, where H2k+1 is the Heisenberg Lie algebra and k≥2. In particular, we prove that g is 3-step solvable, not nilpotent and hence admits no symplectic structure. We show that there exists only one non-abelian Lie algebra of dimension n≤6 which admits a flat Lorentzian metric and a quadratic structure. Using a variant of the double extension method, which conserves both the flat and the quadratic metrics, we prove that any Lie algebra which admits a flat Lorentzian metric and a quadratic structure, is a double extension of an abelian Lie algebra and we construct a large class of such (non-abelian) Lie algebras in higher dimensions. Oscillator Lie algebras denoted by gλ constitute an important class of quadratic Lie algebras which are of the form R⋊H2k+1. It is known that gλ admits no flat Lorentzian metric. We show in this paper that gλ admits a Ricci-flat Lorentzian metric. The paper also study the class of quadratic 2-step nilpotent Lie algebras (g,k) which constitute examples of flat quadratic Lie algebras since k is also flat in this case.

On deformations of the Galilei algebra

In this note we compute all deformations of the 4-dimensional classical Galilei algebra &. In particular, we find examples of quadratic, cubic and quartic Lie algebra deformations.

Weyl quantization of degree 2 symplectic graded manifolds

Let S be a spinor bundle of a pseudo-Euclidean vector bundle (E,g) of even rank. We introduce a new filtration on the algebra D(M,S) of differential operators on S. As a main property, the associated graded algebra grD(M,S) is proved to be isomorphic to the algebra O(M) of smooth functions on M, where M is the degree 2 symplectic graded manifold canonically associated to (E,g). Accordingly, we establish the Weyl quantization of M as a map WQħ:O(M)→D(M,S), and prove that WQħ satisfies all the desired properties of quantizations. As an application, we obtain a bijection between Courant algebroid structures (E,g,ρ,〚⋅,⋅〛) equivalently characterized by Hamiltonian generating functions on M, and skew-symmetric Dirac generating operators D∈D(M,S). The operator D2 gives a new invariant of (E,g,ρ,〚⋅,⋅〛) which generalizes the square norm of the Cartan 3-form of a quadratic Lie algebra. We study this invariant in detail in the particular case of E being the double of a Lie bialgebroid (A,A⁎).

Poisson brackets in Kontsevich’s “Lie World”

In this note we develop the theory of double brackets in the sense of van den Bergh (2008) in Kontsevich’s non-commutative “Lie World”. These double brackets can be thought of as Poisson structures defined by formal expressions only involving the structure maps of a quadratic Lie algebra. The basic example is the Kirillov–Kostant–Souriau (KKS) Poisson bracket.We introduce a notion of non-degenerate double brackets. Surprisingly, in this framework the KKS bracket turns out to be non-degenerate. The main result of the paper is the uniqueness theorem for double brackets with a given moment map. As applications, we establish a monoidal equivalence between Hamiltonian quasi-Poisson spaces and Hamiltonian spaces and give a new proof of the theorem by L. Jeffrey in Jeffrey (1994) on symplectic structure on the moduli space of flat g-connections on a surface of genus 0.

Rota-Baxter operators and non-skew-symmetric solutions of the classical Yang–Baxter equation on quadratic Lie algebras

We study possible connections between Rota-Baxter operators of non-zero weight and non-skew-symmetric solutions of the classical Yang-Baxter equation on finite-dimensional quadratic Lie algebras. The particular attention is made to the case when for a solution $r$ the element $r+\tau(r)$ is $L$-invariant.

Invariant metrics on central extensions of quadratic Lie algebras

A quadratic Lie algebra is a Lie algebra endowed with a symmetric, invariant and nondegenerate bilinear form; such a bilinear form is called an invariant metric. The aim of this work is to describe the general structure of those central extensions of quadratic Lie algebras which in turn have invariant metrics. The structure is such that the central extensions can be described algebraically in terms of the original quadratic Lie algebra, and geometrically in terms of the direct sum decompositions that the invariant metrics involved give rise to.

A minimal-variable symplectic method for isospectral flows

Isospectral flows are abundant in mathematical physics; the rigid body, the the Toda lattice, the Brockett flow, the Heisenberg spin chain, and point vortex dynamics, to mention but a few. Their connection on the one hand with integrable systems and, on the other, with Lie–Poisson systems motivates the research for optimal numerical schemes to solve them. Several works about numerical methods to integrate isospectral flows have produced a large varieties of solutions to this problem. However, many of these algorithms are not intrinsically defined in the space where the equations take place and/or rely on computationally heavy transformations. In the literature, only few examples of numerical methods avoiding these issues are known, for instance, the spherical midpoint method on {{mathfrak {s}}}{{mathfrak {o}}}(3). In this paper we introduce a new minimal-variable, second order, numerical integrator for isospectral flows intrinsically defined on quadratic Lie algebras and symmetric matrices. The algorithm is isospectral for general isospectral flows and Lie–Poisson preserving when the isospectral flow is Hamiltonian. The simplicity of the scheme, together with its structure-preserving properties, makes it a competitive alternative to those already present in literature.

Algebraic loop structures on algebra comultiplications

Abstract In this paper, we study the algebraic loop structures on the set of Lie algebra comultiplications. More specifically, we investigate the fundamental concepts of algebraic loop structures and the set of Lie algebra comultiplications which have inversive, power-associative and Moufang properties depending on the Lie algebra comultiplications up to all the possible quadratic and cubic Lie algebra comultiplications. We also apply those notions to the rational cohomology of Hopf spaces.

Enhanced Leibniz Algebras: Structure Theorem and Induced Lie 2-Algebra

An enhanced Leibniz algebra is an algebraic struture that arises in the context of particular higher gauge theories describing self-interacting gerbes. It consists of a Leibniz algebra $(\mathbb{V},[ \cdot, \cdot ])$, a bilinear form on $\mathbb{V}$ with values in another vector space $\mathbb{W}$, and a map $t \colon \mathbb{W} \to \mathbb{V}$, satisfying altogether four compatibility relations. Our structure theorem asserts that an enhanced Leibniz algebra is uniquely determined by the underlying Leibniz algebra $(\mathbb{V},[ \cdot, \cdot ])$, an appropriate abelian ideal ${\mathfrak i}$ inside it, as well as a cohomology 2-class $[\Delta]$ which only effects the $\mathbb{W}$-valued product. Positive quadratic enhanced Leibniz algebras, as needed for the definition of a Yang-Mills type action functional, turn out to be rather restrictive on the underlying Leibniz algebra $(\mathbb{V},[ \cdot, \dot ])$: $\mathbb{V}$ has to be the hemisemidirect product of a positive quadratic Lie algebra ${\mathfrak g}$ with a ${\mathfrak g}$-module ${\mathfrak i}$, $\mathbb{V} \cong {\mathfrak g}\ltimes{\mathfrak i}$, with ${\mathfrak i}$ the above-mentioned ideal in this case. The second main result of this article is the construction of a functor from the category of such enhanced Leibniz algebras to the category of (semi-strict) Lie 2-algebras or, equivalentely, of two-term $L_\infty$-algebras.