Solvable Lie Algebras Research Articles

Complex structures of splitting type

We study the six-dimensional solvmanifolds that admit complex structures of splitting type classifying the underlying solvable Lie algebras. In particular, many complex structures of this type exist on the Nakamura manifold X , and they allow us to construct a countable family of compact complex non- \partial\bar{\partial} manifolds X_k , k\in\mathbb Z , that admit a small holomorphic deformation \{(X_{k})_{t}\}_{t\in\Delta_k} satisfying the \partial\bar{\partial} -lemma for any t\in\Delta_k except for the central fibre. Moreover, a study of the existence of special Hermitian metrics is also carried out on six-dimensional solvmanifolds with splitting-type complex structures.

Solvable Lie algebras and graphs

We define a solvable extension of the graph 2-step nilpotent Lie algebras of [5] by adding elements corresponding to the 3-cliques of the graph. We study some of their basic properties and we prove that two such Lie algebras are isomorphic if and only if their graphs are isomorphic. We also briefly discuss some metric properties, providing examples of homogeneous spaces with nonpositive curvature operator and solvsolitons.

A Dichotomy for the Gelfand–Kirillov Dimensions of Simple Modules over Simple Differential Rings

The Gelfand–Kirillov dimension has gained importance since its introduction as a tool in the study of non-commutative infinite dimensional algebras and their modules. In this paper we show a dichotomy for the Gelfand–Kirillov dimension of simple modules over certain simple rings of differential operators. We thus answer a question of J. C. McConnell in Representations of solvable Lie algebras V. On the Gelfand-Kirillov dimension of simple modules. McConnell (J. Algebra 76(2), 489–493, 1982) concerning this dimension for a class of algebras that arise as simple homomorphic images of solvable lie algebras. We also determine the Gelfand–Kirillov dimension of an induced module.

Quasi-state rigidity for finite-dimensional Lie algebras

We say that a Lie algebra g is quasi-state rigid if every Ad-invariant continuous Lie quasi-state on it is the directional derivative of a homogeneous quasimorphism. Extending work of Entov and Polterovich, we show that every reductive Lie algebra, as well as the algebras C-n x u( n), n = 1, are rigid. On the other hand, a Lie algebra which surjects onto the three-dimensional Heisenberg algebra is not rigid. For Lie algebras of dimension <= 3 and for solvable Lie algebras which split over a codimension one abelian ideal, we show that this is the only obstruction to rigidity.

Solvability of Equations in Classes of Solvable Groups and Lie Algebras

Solvability of Equations in Classes of Solvable Groups and Lie Algebras

Indecomposable modules of a family of solvable Lie algebras

We classify all uniserial modules of the solvable Lie algebra g=〈x〉⋉V, where V is an abelian Lie algebra over an algebraically closed field of characteristic 0 and x is an arbitrary automorphism of V.

Rigidity-preserving and cohomology-decreasing extensions of solvable rigid Lie algebras

It is shown that the two-known series of rank one and rank two finite-dimensional solvable rigid Lie algebras with non-vanishing second cohomology can be extended to solvable rigid Lie algebras of arbitrary rank such that the cohomology is preserved exactly. For the second series, it is further proved that an extension decreasing the cohomology exists, hence leading to cohomologically rigid Lie algebras.

The Betti Numbers for a Family of Solvable Lie Algebras

We give a property of symplectic quadratic Lie algebras that their Lie algebra of inner derivations has an invertible derivation. A family of symplectic quadratic Lie algebras is introduced to illustrate this situation. Finally, we calculate explicitly the Betti numbers of a family of solvable Lie algebras in two ways: using the cohomology of quadratic Lie algebras and applying a Pouseele’s result on extensions of the one-dimensional Lie algebra by Heisenberg Lie algebras.

Lie Algebras with Nilpotent Length Greater than that of each of their Subalgebras

The main purpose of this paper is to study the finite-dimensional solvable Lie algebras described in its title, which we call minimal non-{mathcal N}. To facilitate this we investigate solvable Lie algebras of nilpotent length k, and of nilpotent length ≤k, and extreme Lie algebras, which have the property that their nilpotent length is equal to the number of conjugacy classes of maximal subalgebras. We characterise the minimal non-{mathcal N} Lie algebras in which every nilpotent subalgebra is abelian, and those of solvability index ≤3.

Solvable extensions of naturally graded quasi-filiform Leibniz algebras of second type ℒ1 and ℒ3

ABSTRACTFor sequences of naturally graded quasi-filiform Leibniz algebras of second type ℒ1 and ℒ3 introduced by Camacho et al., all possible right and left solvable indecomposable extensions over the field ℝ are constructed so that these algebras serve as the nilradicals of the corresponding solvable algebras. The construction continues Winternitz’ and colleagues’ program established to classify solvable Lie algebras using special properties rather than trying to extend one dimension at a time.

Right and left solvable extensions of an associative Leibniz algebra

ABSTRACTA sequence of nilpotent Leibniz algebras denoted by Nn,18 is introduced. Here n denotes the dimension of the algebra defined for n≥4; the first term in the sequence is ℛ18 in the list of four-dimensional nilpotent Leibniz algebras introduced by Albeverio et al. [4]. Then all possible right and left solvable indecomposable extensions over the field ℝ are constructed so that Nn,18 serves as the nilradical of the corresponding solvable algebras. The construction continues Winternitz’ and colleagues’ program established to classify solvable Lie algebras using special properties rather than trying to extend one dimension at a time.

Solvability of a Lie algebra of vector fields implies their integrability by quadratures

We present a substantial generalisation of a classical result by Lie on integrability by quadratures. Namely, we prove that all vector fields in a finite-dimensional transitive and solvable Lie algebra of vector fields on a manifold can be integrated by quadratures.

Minimal faithful upper-triangular matrix representations for solvable Lie algebras

The existence of matrix representations for any given finite-dimensional complex Lie algebra is a classic result on Lie Theory. In particular, such representations can be obtained by means of an isomorphic matrix Lie algebra consisting of upper-triangular square matrices. Unfortunately, there is no general information about the minimal order for the matrices involved in such representations. In this way, our main goal is to revisit, debug and implement an algorithm which provides the minimal order for matrix representations of any finite-dimensional solvable Lie algebra when inserting its law, as well as returning a matrix representative of such an algebra by using the minimal order previously computed. In order to show the applicability of this procedure, we have computed minimal representatives not only for each solvable Lie algebra with dimension less than 6, but also for some solvable Lie algebras of arbitrary dimension.

The generalised nilradical of a Lie algebra

A solvable Lie algebra L has the property that its nilradical N contains its own centraliser. This is interesting because it gives a representation of L as a subalgebra of the derivation algebra of its nilradical with kernel equal to the centre of N. Here we consider several possible generalisations of the nilradical for which this property holds in any Lie algebra. Our main result states that for every Lie algebra L, L/Z(N), where Z(N) is the centre of the nilradical of L, is isomorphic to a subalgebra of Der(N⁎) where N⁎ is an ideal of L such that N⁎/N is the socle of a semisimple Lie algebra.

Negative Ricci curvature on some non-solvable Lie groups

We show that for any non-trivial representation $$(V, \pi )$$ of $$\mathfrak {u}(2)$$ with the center acting as multiples of the identity, the semidirect product $$\mathfrak {u}(2) \ltimes _\pi V$$ admits a metric with negative Ricci curvature that can be explicitly obtained. It is proved that $$\mathfrak {u}(2) \ltimes _\pi V$$ degenerates to a solvable Lie algebra that admits a metric with negative Ricci curvature. An n-dimensional Lie group with compact Levi factor $$\mathrm {SU}(2)$$ admitting a left invariant metric with negative Ricci is therefore obtained for any $$n \ge 7$$ .

A Presentation of the Free Lie Algebra M2,m,3

Let M2,m,3 be a free solvable nilpotent Lie algebra of rank 2 and nilpotency class m - 1. We show that M2,m,3 admits a minimal presentation whose set of defining relators consists of certain types of basic commutators using techniques in Grobner-Shirshov basis theory.

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.

The commutator width of some relatively free lie algebras and nilpotent groups

We determine the exact values of the commutator width of absolutely free and free solvable Lie rings of finite rank, as well as free and free solvable Lie algebras of finite rank over an arbitrary field. We calculate the values of the commutator width of free nilpotent and free metabelian nilpotent Lie algebras of rank 2 or of nilpotency class 2 over an arbitrary field. We also find the values of the commutator width for free nilpotent and free metabelian nilpotent Lie algebras of finite rank at least 3 over an arbitrary field in the case that the nilpotency class exceeds the rank at least by 2. In the case of free nilpotent and free metabelian nilpotent Lie rings of arbitrary finite rank, as well as free nilpotent and free metabelian nilpotent Lie algebras of arbitrary finite rank over the field of rationals, we calculate the values of commutator width without any restrictions. It follows in particular that the free or nonabelian free solvable Lie rings of distinct finite ranks, as well as the free or nonabelian free solvable Lie algebras of distinct finite ranks over an arbitrary field are not elementarily equivalent to each other. We also calculate the exact values of the commutator width of free ℚ-power nilpotent, free nilpotent, free metabelian, and free metabelian nilpotent groups of finite rank.

Fixed Points of Automorphisms Permuting the Generators Cyclically in Free solvable Lie algebras

Fixed Points of Automorphisms Permuting the Generators Cyclically in Free solvable Lie algebras

Solvable Indecomposable Extensions of Two Nilpotent Lie Algebras

A pair of sequences of nilpotent Lie algebras denoted by Nn, 7 and Nn, 16 are introduced. Here, n denotes the dimension of the algebras that are defined for n ≥ 6; the first terms in the sequences are denoted by 6.7 and 6.16, respectively, in the standard list of six-dimensional Lie algebras. For each of them, all possible solvable extensions are constructed so that Nn, 7 and Nn, 16 serve as the nilradical of the corresponding solvable algebras. The construction continues Winternitz’ and colleagues’ program of investigating solvable Lie algebras using special properties rather than trying to extend one dimension at a time.