Perfect Lie Algebras Research Articles

Rota-Baxter operators of non-scalar weights, connections with coboundary Lie bialgebra structures

In the paper, we introduce the notion of a Rota-Baxter operator of a non-scalar weight. As a motivation, we show that there is a natural connection between Rota-Baxter operators of this type and structures of coboundary Lie bialgebras on a quadratic finite-dimensional Lie algebra. We find necessary and sufficient conditions for a pair ( g , δ r ) to be a coboundary (triangular, quasitriangular or factorizable) Lie bialgebra in the case when g is a finite-dimensional quadratic perfect Lie algebra with trivial center. Moreover, we show that some classical results on Lie bialgebras follow from the corresponding results for Rota-Baxter operators.

On the transitivity of Lie ideals and a characterization of perfect Lie algebras

On the transitivity of Lie ideals and a characterization of perfect Lie algebras

Post-Lie algebra structures for perfect Lie algebras

We study the existence of post-Lie algebra structures on pairs of Lie algebras ( g , n ) , where one of the algebras is perfect non-semisimple, and the other one is abelian, nilpotent non-abelian, solvable non-nilpotent, simple, semisimple non-simple, reductive non-semisimple or complete non-perfect. We prove several nonexistence results, but also provide examples in some cases for the existence of a post-Lie algebra structure. Among other results we show that there is no post-Lie algebra structure on ( g , n ) , where g is perfect non-semisimple, and n is s l 3 ( C ) . We also show that there is no post-Lie algebra structure on ( g , n ) , where g is perfect and n is reductive with a 1-dimensional center.

Transposed Poisson structures on solvable and perfect Lie algebras

We described all transposed Poisson algebra structures on oscillator Lie algebras, i.e. on one-dimensional solvable extensions of the -dimensional Heisenberg algebra; on solvable Lie algebras with naturally graded filiform nilpotent radical; on -dimensional solvable extensions of the -dimensional Heisenberg algebra; and on n-dimensional solvable extensions of the n-dimensional algebra with trivial multiplication. We also answered one question on transposed Poisson algebras early posted in a paper by Beites, Ferreira and Kaygorodov. Namely, we found that the semidirect product of and irreducible module gives a finite-dimensional Lie algebra with non-trivial -derivations, but without non-trivial transposed Poisson structures.

On the non existence of sympathetic Lie algebras with dimension less than 25

In this paper, we investigate the question of the lowest possible dimension that a sympathetic Lie algebra [Formula: see text] can attain, when its Levi subalgebra [Formula: see text] is simple. We establish the structure of the nilradical of a perfect Lie algebra [Formula: see text], as a [Formula: see text]-module, and determine the possible Lie algebra structures that one such [Formula: see text] admits. We prove that, as a [Formula: see text]-module, the nilradical must decompose into at least four simple modules. We explicitly calculate the semisimple derivations of a perfect Lie algebra [Formula: see text] with Levi subalgebra [Formula: see text] and give necessary conditions for [Formula: see text] to be a sympathetic Lie algebra in terms of these semisimple derivations. We show that there is no sympathetic Lie algebra of dimension lower than 15, independently of the nilradical’s decomposition. If the nilradical has four simple modules, we show that a sympathetic Lie algebra has dimension greater or equal than 25.

Globalizations of partial group actions on non-associative algebras

This paper is a contribution to the globalization problem for partial group actions on non-associative algebras. We principally focus on partial group actions on Lie algebras, Jordan algebras and Malcev algebras. We give sufficient conditions for the existence and uniqueness of a globalization for partial group actions on the algebras already mentioned. As an application of this result, we show that in characteristic zero every partial group action on a semisimple Malcev algebra admits a globalization, unique up to isomorphism. We give a criterion for the existence and uniqueness of a globalization for a partial group action on a unital Jordan algebra in characteristic different from two, and on a sympathetic Lie algebra (a perfect Lie algebra without center and outer derivations).

Sympathetic Lie algebras and adjoint cohomology for Lie algebras

We study sympathetic (i.e., perfect and complete) Lie algebras. Among other topics they arise in the study of adjoint Lie algebra cohomology. Here a motivation comes from a conjecture of Pirashvili, which says that a finite-dimensional complex perfect Lie algebra is semisimple if and only if its adjoint cohomology vanishes. We prove several general results for sympathetic Lie algebras and for the adjoint Lie algebra cohomology of arbitrary finite-dimensional Lie algebras in characteristic zero using a result of Hochschild and Serre. Moreover, for certain semidirect products we obtain explicit results for the adjoint cohomology.

Complete objects in categories

We introduce the notions of proto-complete, complete, complete ⁎ and strong-complete objects in pointed categories. We show under mild conditions on a pointed exact protomodular category that every proto-complete (respectively complete) object is the product of an abelian proto-complete (respectively complete) object and a strong-complete object. This together with the observation that the trivial group is the only abelian complete group recovers a theorem of Baer classifying complete groups. In addition we generalize several theorems about groups (subgroups) with trivial center (respectively, centralizer), and provide a categorical explanation behind why the derivation algebra of a perfect Lie algebra with trivial center and the automorphism group of a non-abelian (characteristically) simple group are strong-complete.

A Zassenhaus conjecture and CPA-structures on simple modular Lie algebras

Commutative post-Lie algebra structures on Lie algebras, in short CPA structures, have been studied over fields of characteristic zero, in particular for real and complex numbers motivated by geometry. A perfect Lie algebra in characteristic zero only admits the trivial CPA-structure. In this article we study these structures over fields of characteristic p>0. We show that every perfect modular Lie algebra in characteristic p>2 having a solvable outer derivation algebra admits only the trivial CPA-structure. This involves a conjecture by Hans Zassenhaus, saying that the outer derivation algebra Out(g) of a simple modular Lie algebra g is solvable. We try to summarize the known results on the Zassenhaus conjecture and prove some new results using the classification of simple modular Lie algebras by Premet and Strade for algebraically closed fields of characteristic p>3. As a corollary we obtain that every central simple modular Lie algebra of characteristic p>3 admits only the trivial CPA-structure.

Lie triple systems and Leibniz algebras

Abstract The present paper deals with the Lie triple systems via Leibniz algebras. A perfect Lie algebra as a perfect Leibniz algebra and as a perfect Lie triple system is considered and the appropriate universal central extensions are studied. Using properties of Leibniz algebras, it is shown that the Lie triple system universal central extension is either the universal central extension of the Leibniz algebra or the universal central extension of the Lie algebra.

On the equality of triple derivations and derivations of lie algebras

Let L be a Lie algebra over a commutative ring with identity. In the present paper under some mild conditions on L, it is proved that every triple derivation of L is a derivation. In particular, we show that in perfect Lie algebras and free Lie algebras every triple derivation is a derivation. Finally we apply our results to show that every triple derivation of the Lie algebra of block upper triangular matrices is a derivation.

Invariants of contact Lie algebras

The aim of this work is to prove that any contact Lie algebra has a semi-invariant. We explicitly compute it and give conditions for it to be an invariant of the contact Lie algebra. As a consequence we can prove that any perfect contact Lie algebra has trivial center. Moreover, we prove that a perfect Lie algebra is a contact Lie algebra if, and only if, it has only one invariant.

Commutative post-Lie algebra structures on Kac–Moody algebras

We determine commutative post-Lie algebra structures on some infinite-dimensional Lie algebras. We show that all commutative post-Lie algebra structures on loop algebras are trivial. This extends the results for finite-dimensional perfect Lie algebras. Furthermore, we show that all commutative post-Lie algebra structures on affine Kac–Moody Lie algebras are “almost trivial”.

The classification of the perfect cyclic [formula omitted]-modules

A perfect cyclic module of a perfect Lie algebra g with solvable radical r and Levi decomposition g=s⋉r is a finite dimensional module of g generated by an irreducible module of the semisimple Lie algebra s. In this paper we classify all the perfect cyclic finite dimensional indecomposable modules of the perfect Lie algebras sl(n+1)⋉Cn+1, given by the semidirect sum of the simple Lie algebra An with its standard representation. Furthermore, using the embedding of the Lie algebra sl(n+1)⋉Cn+1 in sl(n+2), we show that any finite dimensional irreducible module of sl(n+2) restricted to sl(n+1)⋉Cn+1 is a perfect cyclic module and that any perfect cyclic sl(n+1)⋉Cn+1-module can be constructed as quotient module of the restriction to sl(n+1)⋉Cn+1 of some finite dimensional irreducible sl(n+2)-module. This explicit realization of the perfect cyclic sl(n+1)⋉Cn+1-modules plays a role in their classification.

Commutative post-Lie algebra structures on Lie algebras

We show that any CPA-structure (commutative post-Lie algebra structure) on a perfect Lie algebra is trivial. Furthermore we give a general decomposition of inner CPA-structures, and classify all CPA-structures on complete Lie algebras. As a special case we obtain the CPA-structures of parabolic subalgebras of semisimple Lie algebras.

Post-Lie algebra structures on pairs of Lie algebras

We study post-Lie algebra structures on pairs of Lie algebras (g,n), which describe simply transitive nil-affine actions of Lie groups. We prove existence results for such structures depending on the interplay of the algebraic structures of g and n. We consider the classes of simple, semisimple, reductive, perfect, solvable, nilpotent, abelian and unimodular Lie algebras. Furthermore we consider commutative post-Lie algebra structures on perfect Lie algebras. Using Lie algebra cohomology we can classify such structures in several cases. We also study commutative structures on low-dimensional Lie algebras and on nilpotent Lie algebras.

Bicrossed Products, Matched Pair Deformations and the Factorization Index for Lie Algebras

For a perfect Lie algebra $\mathfrak{h}$ we classify all Lie algebras containing $\mathfrak{h}$ as a subalgebra of codimension $1$. The automorphism groups of such Lie algebras are fully determined as subgroups of the semidirect product $\mathfrak{h} \ltimes (k^* \times {\rm Aut}_{\rm Lie} (\mathfrak{h}))$. In the non-perfect case the classification of these Lie algebras is a difficult task. Let $\mathfrak{l} (2n+1, k)$ be the Lie algebra with the bracket $[E_i, G] = E_i$, $[G, F_i] = F_i$, for all $i = 1, \dots, n$. We explicitly describe all Lie algebras containing $\mathfrak{l} (2n+1, k)$ as a subalgebra of codimension $1$ by computing all possible bicrossed products $k \bowtie \mathfrak{l} (2n+1, k)$. They are parameterized by a set of matrices ${\rm M}_n (k)^4 \times k^{2n+2}$ which are explicitly determined. Several matched pair deformations of $\mathfrak{l} (2n+1, k)$ are described in order to compute the factorization index of some extensions of the type $k \subset k \bowtie \mathfrak{l} (2n+1, k)$. We provide an example of such extension having an infinite factorization index.

Triple Homomorphisms of Perfect Lie Algebras

Let L, L′ be Lie algebras over a commutative ring R. A R-linear mapping f: L → L′ is called a triple homomorphism from L to L′ if f([x, [y, z]]) = [f(x), [f(y), f(z)]] for all x, y, z ∈ L. It is clear that homomorphisms, anti-homomorphisms, and sums of homomorphisms and anti-homomorphisms are all triple homomorphisms. We proved that, under certain assumptions, these are all triple homomorphisms.

Triple Derivations of Perfect Lie Algebras

We investigate the triple derivation algebras of Lie algebras. It is proved that, if the base ring contains , L is a perfect Lie algebra with zero center, then every triple derivation of L is a derivation, and every triple derivation of the derivation algebra Der(L) is an inner derivation.

On Strongly Perfect Lie Algebras

Using higher Leibniz homology, we introduce the notions of k-perfect and strongly perfect Lie algebras and discuss a relationship with semisimple Lie algebras.