Left-symmetric Algebras Research Articles

Left-symmetric algebra structures on the finite-dimensional Witt algebra

We establish compatible left-symmetric algebra structures on the finite-dimensional Witt algebra over an algebraically closed field of characteristic p>3, satisfying specific Z or Zp-graded conditions. Our primary method involves the representation theory of the Witt algebra in Zp-graded case. Additionally, as a consequential result, we partially answer a question raised by Shen.

Complete descriptions of pseudo-Euclidean left-symmetric L-algebras and their pseudo-Euclidean modules

A pseudo-Euclidean left-symmetric algebra (A,.,〈,〉) is a left-symmetric algebra endowed with a non-degenerate symmetric bilinear form 〈,〉 such that left multiplications by any element of A are skew-symmetric with respect to 〈,〉. We recall that a pseudo-Euclidean Lie algebra (g,[,],〈,〉) is flat if and only if (g,.,〈,〉), its underlying vector space endowed with the Levi-Civita product associated with 〈,〉, is a pseudo-Euclidean left-symmetric algebra. In this paper, we study pseudo-Euclidean left-symmetric algebras (A,.,〈,〉) such that commutators of all elements of A are contained in the left annihilator of (A,.), these algebras will be called pseudo-Euclidean left-symmetric L-algebras. Next, we introduce and study pseudo-Euclidean modules of pseudo-Euclidean left-symmetric L-algebras. We develop double extension processes that allow us to have inductive descriptions of all pseudo-Euclidean left-symmetric L-algebras and of all its pseudo-Euclidean modules of any signature. This, in particular, improves some of the results obtained in [11].

Abelian Extensions of Modified λ-Differential Left-Symmetric Algebras and Crossed Modules

In this paper, we define a cohomology theory of a modified λ-differential left-symmetric algebra. Moreover, we introduce the notion of modified λ-differential left-symmetric 2-algebras, which is the categorization of a modified λ-differential left-symmetric algebra. As applications of cohomology, we classify linear deformations and abelian extensions of modified λ-differential left-symmetric algebras using the second cohomology group and classify skeletal modified λ-differential left-symmetric 2-algebra using the third cohomology group. Finally, we show that strict modified λ-differential left-symmetric 2-algebras are equivalent to crossed modules of modified λ-differential left-symmetric algebras.

Left-Symmetric Superalgebra Structures on an Infinite-Dimensional Lie Superalgebra

In this note, the compatible left-symmetric superalgebra structures on an infinite-dimensional Lie superalgebra with some natural grading conditions are completely determined. The results of earlier work on left-symmetric algebra structures on the Virasoro algebra play an essential role in determining these compatible structures.

Left-symmetric superalgebras on special linear Lie superalgebras

In this paper, we study the existence and classification problems of left-symmetric superalgebras on special linear Lie superalgebras sl(m|n) with m≠n. The main three results of this paper are: (i) a complete classification of the left-symmetric superalgebras on sl(2|1), (ii) sl(m|1) does not admit left-symmetric superalgebras for m≥3, and (iii) sl(m+1|m) admits a left-symmetric superalgebra for every m≥1. To prove these results we combine previous results on the existence and classification of left-symmetric algebras on the Lie algebras glm with a detailed analysis of small representations of the Lie superalgebras sl(m|1). We also conjecture that sl(m|n) admits left-symmetric superalgebras if and only if m=n+1.

Left-symmetric algebra structures on contact Lie algebras

The aim of this work is to give necessary and sufficient conditions to extend an LSA structure on a Lie algebra to a Lie algebra that either contains as a subalgebra, or is a central extension of it. We also study conditions under which a Lie algebra together with an LSA structure admit a Frobenius or a contact form. Furthermore, we describe the properties of the Reeb vector associated with the codimension one contact ideal with respect to the LSA structure on the Frobenius Lie algebra.

Affine automorphisms of the universal multiplicative enveloping algebra of the two-dimensional left-symmetric algebra with zero multiplication

We construct the basis of the universal multiplicative enveloping algebra of the left-symmetric algebra. We also describe the affine automorphisms of the universal multiplicative enveloping algebra of the twodimensional left-symmetric algebra with zero multiplication.

A Left-Symmetric Structure on The Semi-Direct Sum Real Frobenius Lie Algebra of Dimension 8

Let be the Lie algebra of the semi-direct sum of the real vector space and the Lie algebra of the sets of all real matrices. In this paper, a Frobenius functional is constructed in order for the Lie algebra to be the real Frobenius Lie algebra of dimension 8. Moreover, a bilinear form corresponding to this Frobenius functional is symplectic. Then the obtained symplectic bilinear form induces the left-symmetric algebra structures on . In other words, the Lie algebra is the left-symmetric algebra. In particular, we give the formulas of its left-symmetric algebra structure explicitely. The left-symmetric algebra structures for case of higher dimension of this Lie algebra type are still an open problem to be investigated.

On the embedding of left-symmetric algebras into differential Perm-algebras

Given an associative algebra satisfying the left commutativity identity abc = bac (Perm-algebra) with a derivation d, the new operation is left-symmetric (pre-Lie). We derive necessary and sufficient conditions for a left-symmetric algebra to be embeddable into a differential Perm-algebra.

The algebraic and geometric classification of nilpotent left-symmetric algebras

This paper is devoted to the complete algebraic and geometric classification of complex 4-dimensional nilpotent left-symmetric algebras. The corresponding geometric variety has dimension 15 and decomposes into 3 irreducible components determined by the Zariski closures of two one-parameter families of algebras and a two-parameter family of algebras (see Theorem B). In particular, there are no rigid 4-dimensional complex nilpotent left symmetric algebras.

On k-para-Kähler Lie algebras, a subclass of k-symplectic Lie algebras

k-Para-Kähler Lie algebras are a generalization of para-Kähler Lie algebras (k = 1) and constitute a subclass of k-symplectic Lie algebras. In this paper, we show that the characterization of para-Kähler Lie algebras as left symmetric bialgebras can be generalized to k-para-Kähler Lie algebras leading to the introduction of two new structures which are different but both generalize the notion of left symmetric algebra. This permits also the introduction of generalized S-matrices. We determine then all the k-symplectic Lie algebras of dimension and all the six dimensional 2-para-Kähler Lie algebras.

Pre-anti-flexible bialgebras

In this paper, we derive pre-anti-flexible algebras structures in term of zero weight’s Rota-Baxter operators, built underlying left-symmetric algebras, view pre-anti-flexible algebras as a splitting of anti-flexible algebras, introduce pre-anti-flexible bialgebras and establish its equivalences among matched pair of anti-flexible algebras and matched pair of pre-anti-flexible algebras. Special class of pre-anti-flexible bialgebras leads to the establishment of the pre-anti-flexible Yang-Baxter equation which is the same with -equation. Symmetric solution of pre-anti-flexible Yang-Baxter equation gives a pre-anti-flexible bialgebra. Finally, we recall and link -operators of anti-flexible algebras to bimodules of pre-anti-flexible algebras and built symmetric solutions of anti-flexible Yang-Baxter equation.

Nijenhuis geometry II: Left-symmetric algebras and linearization problem for Nijenhuis operators

A field of endomorphisms R is called a Nijenhuis operator if its Nijenhuis torsion vanishes. In this work we study a specific kind of singular points of R called points of scalar type. We show that the tangent space at such points possesses a natural structure of a left-symmetric algebra (also known as pre-Lie or Vinberg-Kozul algebras). Following Weinstein's approach to linearization of Poisson structures, we state the linearisation problem for Nijenhuis operators and give an answer in terms of non-degenerate left-symmetric algebras. In particular, in dimension 2, we give classification of non-degenerate left-symmetric algebras for the smooth category and, with some small gaps, for the analytic one. These two cases, analytic and smooth, differ. We also obtain a complete classification of two-dimensional real left-symmetric algebras, which may be an interesting result on its own.

On pseudo-Euclidean Novikov algebras

A flat pseudo-Euclidean Lie algebra is a left-symmetric algebra endowed with a non-degenerate symmetric bilinear form such that left multiplications are skew-symmetric. We observe that, in many classes of flat pseudo-Euclidean Lie algebras, the left-symmetric product is actually of Novikov. This leads us to study pseudo-Euclidean Novikov algebras (A,〈,〉), that is, Novikov algebra A endowed with a non-degenerate symmetric bilinear form such that left multiplications are skew-symmetric. We show that a Lorentzian Novikov algebra (A,〈,〉) must be transitive. This implies that the underlying Lie algebra AL must be unimodular. In geometrical terms, the left-invariant metric on a corresponding Lie group is geodesically complete. In this case, we show that the restriction of the product to [AL,AL]⊥ is trivial. Using the double extension process, we prove that (A,〈,〉) is a pseudo-Euclidean Novikov algebra such that [AL,AL] is Lorentzian if and only if AL splits as AL=[AL,AL]⊥⊕[AL,AL] where [AL,AL] and [AL,AL]⊥ are abelian, [AL,AL] is Lorentzian and adx is skew-symmetric for any x∈[AL,AL]⊥. This also implies, in this case, that A is transitive, AL is unimodular and two-solvable. This result solves the problem for Lorentzian Novikov algebras such that [AL,AL] is non-degenerate. If [AL,AL] is degenerate, we show that such Lorentzian Novikov algebra is obtained by the double extension process from a flat Euclidean algebra, and we give some applications in low dimensions.

Vaisman solvmanifolds and relations with other geometric structures

Fil: Andrada, Adrian Marcelo. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Centro Cientifico Tecnologico Conicet - Cordoba. Centro de Investigacion y Estudios de Matematica. Universidad Nacional de Cordoba. Centro de Investigacion y Estudios de Matematica; Argentina. Universidad Nacional de Cordoba. Facultad de Matematica, Astronomia y Fisica; Argentina

Left-Symmetric Algebra Structures on the Planar Galilean Conformal Algebra

In this paper, we mainly determine the compatible left-symmetric algebra structures on the planar Galilean conformal algebra with some natural grading conditions. The results of earlier work on left-symmetric algebra structures on the twisted Heisenberg-Virasoro algebra play an important role in determining these compatible structures. As a corollary, any such left-symmetric algebra contains an infinite-dimensional non-trivial subalgebra that is also a submodule of the regular module.

Chain of squares and nilpotency of left symmetric algebras

The main result of this paper is the following: we show that if for a left symmetric algebra there exists a finite strictly ascending chain (or sequence) of two side ideals and the square of each element of an ideal is in the preceding one, then the algebra is nilpotent.

Rota-Baxter operators of 3-dimensional Heisenberg Lie algebra

In this paper, we consider the question of the Rota-Baxter operators of 3-dimensional Heisenberg Lie algebra on $\mathbb{F}$, where $\mathbb{F}$ is an algebraic closed field. By using the Lie product of the basis elements of Heisenberg Lie algebras, all Rota-Baxter operators of 3-dimensional Heisenberg Lie algebras are calculated and left symmetric algebras of 3-dimensional Heisenberg Lie algebra are determined by using the Yang-Baxter operators.

Classification of derivation algebras in low dimensions

A left-symmetric algebra A is said to be a derivation algebra if all its left and right multiplications are derivations of the Lie algebra associated to A. In this paper, we shall study derivation algebras over R and then classify almost all those of dimensions ≤4.

Hyper-para-Kähler Lie algebras with abelian complex structures and their classification up to dimension 8

Hyper-para-Kahler structures on Lie algebras where the complex structure is abelian are studied. We show that there is a one-to-one correspondence between such hyper-para-Kahler Lie algebras and complex commutative (hence, associative) symplectic left-symmetric algebras admitting a semilinear map $$K_s$$ verifying certain algebraic properties. Such equivalence allows us to give a complete classification, up to holomorphic isomorphism, of pairs $$({\mathfrak g},J)$$ of 8-dimensional Lie algebras endowed with abelian complex structures which admit hyper-para-Kahler structures.