One-dimensional Central Extension Research Articles

Extensions of solvable Lie algebras with naturally graded filiform nilradical

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].

Central extensions of 2-filiform Leibniz algebras

In this paper we describe the central extensions of some nilpotent Leibniz algebras and one-dimensional central extensions (up to isomorphism) of all naturally graded 2-filiform Leibniz algebras over the field of complex numbers. It is well-know that 2-filiform Leibniz algebras which are not naturally graded have not yet studied. There are such kind of algebras in the algebras obtained along this article. Also in this work we describe some subclasses of 6-dimensional nilpotent Leibniz algebras with condition quotient algebra by two-dimensional center of the algebra is isomorphic to the four-dimensional naturally graded 2-filiform nonsplit Leibniz algebra.

Invariant Conformal Killing–Yano 2-Forms on Five-Dimensional Lie Groups

We study left invariant conformal Killing–Yano (CKY) 2-forms on Lie groups endowed with a left invariant metric. We classify all 5-dimensional metric Lie algebras carrying CKY tensors that are obtained as a one-dimensional central extension of 4-dimensional metric Lie algebras endowed with an invertible parallel skew-symmetric tensor. On the other hand, we also classify 5-dimensional metric Lie algebras with center of dimension greater than one admitting strict CKY tensors. In addition, we determine all possible CKY tensors on these metric Lie algebras. In particular, we exhibit the first examples of CKY 2-forms on metric Lie algebras which do not admit any Sasakian structure.

Central Extension of Left Alternative Algebras and the Second Cohomology Group

The purpose of this paper is to prove that the second cohomology group H 2 A , F of a left alternative algebra A over an algebraically closed field F of characteristic 0 can be interpreted as the set of equivalent classes of one-dimensional central extensions of A .

A class of nilpotent Lie algebras whose center acts nontrivially in cohomology

We show that the central representation is nontrivial for all one-dimensional central extensions of nilpotent Lie algebras possessing a codimension one abelian ideal.

Cohomology of Restricted Filiform Lie Algebras m2λ(p)

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 there is a family ${\mathfrak m}_2^{\lambda}(p)$ of restricted Lie algebra structures parameterized by elements $\lambda \in {\mathbb F}^p$. We explicitly describe bases for the ordinary and restricted 1- and 2-cohomology spaces with trivial coefficients, and give formulas for the bracket and $[p]$-operations in the corresponding restricted one-dimensional central extensions.

Quadratic Leibniz conformal algebras

In this paper, we study a class of Leibniz conformal algebras called quadratic Leibniz conformal algebras. An equivalent characterization of a Leibniz conformal algebra [Formula: see text] through three algebraic operations on [Formula: see text] is given. By this characterization, several constructions of quadratic Leibniz conformal algebras are presented. Moreover, one-dimensional central extensions of quadratic Leibniz conformal algebras are considered using some bilinear forms on [Formula: see text]. In particular, we also study one-dimensional Leibniz central extensions of quadratic Lie conformal algebras.

Restricted one-dimensional central extensions of the restricted filiform Lie algebras [formula omitted

Consider the filiform Lie algebra m0 with nonzero Lie brackets [e1,ei]=ei+1 for 1<i<p, where the characteristic of the field F is p>0. We show that there is a family m0λ(p) of restricted Lie algebra structures parameterized by elements λ∈Fp. We explicitly describe both the ordinary and restricted 1-cohomology spaces and show that for p≥3 these spaces are equal. We also describe the ordinary and restricted 2-cohomology spaces and interpret our results in the context of one-dimensional central extensions.

Extended noncommutative Minkowski spacetimes and hybrid gauge symmetries

We study the Lie bialgebra structures that can be built on the one-dimensional central extension of the Poincaré and (A)dS algebras in (1 + 1) dimensions. These central extensions admit more than one interpretation, but the simplest one is that they describe the symmetries of (the noncommutative deformation of) an Abelian gauge theory, U(1) or SO(2) on Minkowski or (A)dS spacetime. We show that this highlights the possibility that the algebra of functions on the gauge bundle becomes noncommutative. This is a new way in which the Coleman–Mandula theorem could be circumvented by noncommutative structures, and it is related to a mixing of spacetime and gauge symmetry generators when they act on tensor-product states. We obtain all Lie bialgebra structures on centrally-extended Poincaré and (A)dS which are coisotropic w.r.t. the Lorentz algebra, and therefore admit the construction of a noncommutative principal gauge bundle on a quantum homogeneous spacetime. It is shown that several different types of hybrid noncommutativity between the spacetime and gauge coordinates are allowed. In one of these cases, an alternative interpretation of the central extension leads to a new description of the well-known canonical noncommutative spacetime as the quantum homogeneous space of a quantum Poincaré algebra.

Modules induced from polynomial subalgebras of the Virasoso algebra

The Virasoro Lie algebra is a one-dimensional central extension of the Witt algebra, which can be realized as the Lie algebra of derivations on the algebra C[t±] of Laurent polynomials. Using this fact, we define a natural family of subalgebras of the Virasoro algebra, which we call polynomial subalgebras. We describe the one-dimensional modules for polynomial subalgebras, and we use this description to study the corresponding induced modules for the Virasoro algebra. We show that these induced modules are frequently simple and generalize a family of recently discovered simple modules. Additionally, we explore tensor products involving these induced modules. This allows us to describe new simple modules and also recover results on recently discovered simple modules for the Virasoro algebra.

Q-Deformations of 3-Lie Algebras

q-Deformations of 3-Lie algebras and representations of q-3-Lie algebras are discussed. A q-3-Lie algebra [Formula: see text], where [Formula: see text] and [Formula: see text], is a vector space A over a field 𝔽 with 3-ary linear multiplications [ , , ]q and [Formula: see text] from [Formula: see text] to A, and a map [Formula: see text] satisfying the q-Jacobi identity [Formula: see text] for all [Formula: see text]. If the multiplications satisfy that [Formula: see text] and [Formula: see text] is skew-symmetry, then [Formula: see text] is called a type (I)-q-3- Lie algebra. From q-Lie algebras, group algebras and commutative associative algebras, q-3-Lie algebras and type (I)-q-3-Lie algebras are constructed. Also, the non-trivial onedimensional central extension of q-3-Lie algebras is investigated, and new q-3-Lie algebras [Formula: see text], and [Formula: see text] are obtained.

Restricted one-dimensional central extensions of restricted simple Lie algebras

We study the restricted one-dimensional central extensions of an arbitrary finite dimensional restricted simple Lie algebra for p≥5. For H2(g)=0, we explicitly describe the cocycles spanning H⁎2(g), and in the case H2(g)≠0, we give a procedure to describe a basis for H⁎2(g).

Restricted cohomology of modular Witt algebras

Let p > 3 be a prime, K be an algebraically closed field of characteristic p and A = K[x]/(x − 1). The Lie algebra W = Der(A) is called the modular Witt algebra. It has a basis ei = x ∂ where i = −1, 0, . . . , p − 2 and ∂ = d/dx. The Lie bracket in W satisfies [ei, ej] = (j − i)ei+j where i + j is computed modulo p. In characteristic zero, the analogous Lie algebra of derivations of the algebra C[t, t] is also called the Witt algebra. Gelfand and Fuchs have shown that Der(C[t, t]) has, up to equivalence, exactly one non-trivial onedimensional central extension (the Virasoro algebra) [5]. Block has shown in characteristic p > 5 that the modular Witt algebra also has, again up to equivalence, exactly one non-trivial one-dimensional central extension as an ordinary (i.e. non-restricted) Lie algebra [1].

Nontrivial central extensions of 3-Lie algebras

We construct a series of infinite-dimensional 3-Lie algebras which include w∞ and SDiff(T2) 3-Lie algebras as subalgebras, and determine their nontrivial one-dimensional central extensions.

Magnetic Geodesic Flows on Homogeneous Manifolds

The algebra of the integrals of motion, linear in the momenta, of magnetic geodesic flows on arbitrary Riemannian manifolds is investigated. It is shown that the indicated algebra is a one-dimensional central extension of the Lie algebra of Killing vector fields of the manifold which conserve the external field. A constructive method is proposed for performing an integration in quadratures of the magnetic geodesic flows on homogeneous Riemannian manifolds of zero index.

Low dimensional cohomology of Hom-Lie algebras and q-deformed W(2, 2) algebra

This paper aims to study low dimensional cohomology of Hom-Lie algebras and the q-deformed W(2, 2) algebra. We show that the q-deformed W(2, 2) algebra is a Hom-Lie algebra. Also, we establish a one-to-one correspondence between the equivalence classes of one-dimensional central extensions of a Hom-Lie algebra and its second cohomology group, leading us to determine the second cohomology group of the q-deformed W(2, 2) algebra. In addition, we generalize some results of derivations of finitely generated Lie algebras with values in graded modules to Hom-Lie algebras. As application, we compute all αk-derivations and in particular the first cohomology group of the q-deformed W(2, 2) algebra.

Integrating Klein-Gordon-Fock equations in an external electromagnetic field on Lie groups

We investigate the structure of the Klein-Gordon-Fock equation symmetry algebra on pseudo-Riemannian manifolds with motions in the presence of an external electromagnetic field. We show that in the case of an invariant electromagnetic field tensor, this algebra is a one-dimensional central extension of the Lie algebra of the group of motions. Based on the coadjoint orbit method and harmonic analysis on Lie groups, we propose a method for integrating the Klein-Gordon-Fock equation in an external field on manifolds with simply transitive group actions. We consider a nontrivial example on the four-dimensional group $E(2) \times \mathbb{R}$ in detail.

Low-Dimensional Cohomology Groups of the Lie Algebras W(a, b)

In this article, we determine one-dimensional (Leibniz) central extensions, the derivation algebra, and the automorphism group of the infinite-dimensional Lie algebra W(a, b) (for a, b ∈ ℂ) which is the semidirect product Lie algebra of the Witt algebra and its tensor density module I(a, b).

Owner-reported breed-typical behavior and breed-group differences in the german pet dog population

We study the restricted one-dimensional central extensions of an arbitrary finite dimensional restricted simple Lie algebra for p≥5. For H2(g)=0, we explicitly describe the cocycles spanning H⁎2(g), and in the case H2(g)≠0, we give a procedure to describe a basis for H⁎2(g).

Verma Modules over Quantum Torus Lie Algebras

AbstractRepresentations of various one-dimensional central extensions of quantum tori (called quantum torus Lie algebras) were studied by several authors. Now we define a central extension of quantum tori so that all known representations can be regarded as representations of the new quantum torus Lie algebras . The center of now is generally infinite dimensional.In this paper, Z-graded Verma modules over and their corresponding irreducible highest weight modules are defined for some linear functions . Necessary and sufficient conditions for to have all finite dimensional weight spaces are given. Also necessary and sufficient conditions for Verma modules e to be irreducible are obtained.