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AbstractRecently we have investigated Lie algebras and abelian Lie algebras derived from Lie hyperalgebras using the fundamental relations L and A, respectively. In the present paper, continuing this method we obtain solvable Lie algebras from Lie hyperalgebras by Sn-relations. We show that ∩n≥1Sn* is the smallest equivalence relation on a Lie hyperalgebra such that the quotient structure is a solvable Lie algebra. We also provide some necessary and sufficient conditions for transitivity of the relation Sn using the notion of Sn-part. [ABSTRACT FROM AUTHOR]

In this paper, we study strong Kähler with torsion (SKT) and generalized Kähler structures on solvable Lie algebras with (not necessarily abelian) codimension 2 nilradical. We treat separately the case of J$J$‐invariant nilradical and non‐J$J$‐invariant nilradical. A classification of such SKT Lie algebras in dimension 6 is provided. In particular, we give a general construction to extend SKT nilpotent Lie algebras to SKT solvable Lie algebras of higher dimension, and we construct new examples of SKT and generalized Kähler compact solvmanifolds. [ABSTRACT FROM AUTHOR]

We determine the structure of solvable Lie groups endowed with invariant stretched non-positive Weyl connections and find classes of solvable Lie groups admitting and not admitting such connections. In dimension 4 we fully classify solvable Lie groups which admit invariant SNP Weyl connections. [ABSTRACT FROM AUTHOR]

The aim of the paper is to provide a characterization criterion of exponential solvable Frobenius Lie algebras (having open coadjoint orbits), in terms of primitive ideals of the associated enveloping algebra. In the case of complex solvable Lie algebras, we also show that an algebraic adjoint orbit is open if and only if the associated primitive ideal through the Dixmier map is trivial. [ABSTRACT FROM AUTHOR]

A constructive procedure is given to determine all ideals of a finite-dimensional solvable Lie algebra. This is used in determining all conjugacy classes of subalgebras of a given finite-dimensional solvable Lie algebra. [ABSTRACT FROM AUTHOR]

We define a four-dimensional Lie algebra g in this paper and then prove that this Lie algebra is solvable but not nilpotent. Due to the fact that g is a Lie algebra, ∀ x , y ∈ g , [ x , y ] = − [ y , x ] , that is, the operation [ , ] has anti symmetry. Symmetry is a very important law, and antisymmetry is also a very important law. We studied the structure of Poisson algebras on g using the matrix method. We studied the necessary and sufficient conditions for the automorphism of this class of Lie algebras, and give the decomposition of its automorphism group by A u t (g) = G 3 G 1 G 2 G 3 G 4 G 7 G 8 G 5 , or A u t (g) = G 3 G 1 G 2 G 3 G 4 G 7 G 8 G 5 G 6 , or A u t (g) = G 3 G 1 G 2 G 3 G 4 G 7 G 8 G 5 G 3 , where G i is a commutative subgroup of A u t (g) . We give some subgroups of g's automorphism group and systematically studied the properties of these subgroups. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce a new class of locally conformal Kähler manifolds which will generalize the Vaisman manifold. Then, some basic properties of this class is discussed, also the existence of such manifolds is shown with concrete examples. As an application, we study such structures on four-dimensional solvable Lie algebra. [ABSTRACT FROM AUTHOR]

In mathematics and engineering, a manifold is a topological space that locally resembles Euclidean space near each point. Defining the best metric for these manifolds have several engineering and science implications from controls to optimization for generalized inner product applications of Gram Matrices that appear in these applications. These smooth geometric manifold applications can be formalized by Lie Groups and their Lie Algebras on its infinitesimal elements. Nilpotent matrices that are matrices with zero power with left-invariant metric on Lie group with non-commutative properties namely non-abelian nilsoliton metric Lie algebras will be the focus of this article. In this study, we present an algorithm to classify eigenvalues of nilsoliton derivations for 9-D non-abelian nilsoliton metric Lie algebras with non-singular Gram matrices. [ABSTRACT FROM AUTHOR]

Extensions of finite-dimensional nilpotent Lie algebras, in particular, solvable extensions, are considered. Some properties of maximal extensions are proved. A counterexample to L. Šnobl's conjecture concerning the uniqueness of maximal solvable extensions is constructed. [ABSTRACT FROM AUTHOR]

There are many routines developed for solving ordinary differential Equations (ODEs) of different types. In the case of an nth-order ODE that admits an r-parameter Lie group (3 ≤ r ≤ n) , there is a powerful method of Lie symmetry analysis by which the ODE is reduced to an (n − r) th-order ODE plus r quadratures provided that the Lie algebra formed by the infinitesimal generators of the group is solvable. It would seem this method is not widely appreciated and/or used as it is not mentioned in many related articles centred around integration of higher order ODEs. In the interest of mainstreaming the method, we describe the method in detail and provide four illustrative examples. We use the case of a third-order ODE that admits a three-dimensional solvable Lie algebra to present the gist of the integration algorithm. [ABSTRACT FROM AUTHOR]

We define a four-dimensional Lie algebra g in this paper and then prove that this Lie algebra is solvable but not nilpotent. Due to the fact that g is a Lie algebra, ∀x,y∈g,[x,y]=−[y,x], that is, the operation [,] has anti symmetry. Symmetry is a very important law, and antisymmetry is also a very important law. We studied the structure of Poisson algebras on g using the matrix method. We studied the necessary and sufficient conditions for the automorphism of this class of Lie algebras, and give the decomposition of its automorphism group by Aut(g)=G3G1G2G3G4G7G8G5, or Aut(g)=G3G1G2G3G4G7G8G5G6, or Aut(g)=G3G1G2G3G4G7G8G5G3, where Gi is a commutative subgroup of Aut(g). We give some subgroups of g’s automorphism group and systematically studied the properties of these subgroups.

Given a nilpotent Lie algebra, we study the space of all diagonalizable derivations such that the corresponding one-dimensional solvable extension admits a left-invariant metric with negative Ricci curvature. Lauret and Will have conjectured that such a space coincides with an open and convex subset of derivations defined in terms of the moment map for the variety of nilpotent Lie algebras. We prove the validity of the conjecture in dimensions ≤ 5, as well as for Heisenberg Lie algebras and standard filiform Lie algebras. [ABSTRACT FROM AUTHOR]

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There are many routines developed for solving ordinary differential Equations (ODEs) of different types. In the case of an nth-order ODE that admits an r-parameter Lie group (3≤r≤n), there is a powerful method of Lie symmetry analysis by which the ODE is reduced to an (n−r)th-order ODE plus r quadratures provided that the Lie algebra formed by the infinitesimal generators of the group is solvable. It would seem this method is not widely appreciated and/or used as it is not mentioned in many related articles centred around integration of higher order ODEs. In the interest of mainstreaming the method, we describe the method in detail and provide four illustrative examples. We use the case of a third-order ODE that admits a three-dimensional solvable Lie algebra to present the gist of the integration algorithm.

We show that in the class of solvable Lie algebras there exist algebras which admit local derivations which are not ordinary derivation and also algebras for which every local derivation is a derivation. We found necessary and sufficient conditions under which any local derivation of solvable Lie algebras with abelian nilradical and one-dimensional complementary space is a derivation. Moreover, we prove that every local derivation on a finite-dimensional solvable Lie algebra with model nilradical and maximal dimension of complementary space is a derivation. [ABSTRACT FROM AUTHOR]

We classify and investigate locally conformally Kähler structures on four-dimensional solvable Lie algebras up to linear equivalence. As an application we can produce many examples in higher dimension, here including lcK structures on Oeljeklaus-Toma manifolds, and we also give a geometric interpretation of some of the 4-dimensional structures in our classification.

Let P be a Poisson algebra with a Lie bracket { , } over a field F of characteristic p ≥ 0. In this paper, the Lie structure of P is investigated. In particular, if P is solvable with respect to its Lie bracket, then we prove that the Poisson ideal J of P generated by all elements { { { x 1 , x 2 } , { x 3 , x 4 } } , x 5 } with x 1 , ... , x 5 ∈ P is associative nilpotent of index bounded by a function of the derived length of P. We use this result to further prove that if P is solvable and p ≠ 2 , then the Poisson ideal { P , P } P is nil. [ABSTRACT FROM AUTHOR]

Let F be a field of algebraically closed and L be a finite dimensional Lie algebra over field F . L ′ = [ L , L ] denotes the derived subalgebra of L. Following the analogy with group theory, we define the subalgebra D(L) of L to be the intersection of the normalizer of the derived subalgebras of all subalgebras of L. In a Lie algebra L, this is an ideal of L, allowing the definition of the ascending series: set D 0 (L) = 0 , D i + 1 (L) / D i (L) = D (L / D i (L)) for i ≥ 1 , D ∞ (L) denotes the terminal term of the ascending series. It is proved that L is solvable if and only if L = D ∞ (L) . [ABSTRACT FROM AUTHOR]

Let L be a Lie algebra over a field of positive characteristic and let S (L) and s (L) denote, respectively, the symmetric Poisson algebra and the truncated symmetric Poisson algebra of L. As a natural continuation of the work by Monteiro Alves and Petrogradsky in [9] , we investigate the structure of L when S (L) or s (L) is solvable. We first disprove a conjecture stated in [9] about solvability of S (L) (and s (L)) in characteristic 2. Next, the derived lengths of s (L) are studied. In particular, we provide bounds for the derived lengths of s (L) , establish when s (L) is metabelian, and characterize truncated symmetric Poisson algebras of minimal derived length. [ABSTRACT FROM AUTHOR]

We classify and investigate locally conformally Kähler structures on four-dimensional solvable Lie algebras up to linear equivalence. As an application we can produce many examples in higher dimension, here including lcK structures on Oeljeklaus-Toma manifolds, and we also give a geometric interpretation of some of the 4-dimensional structures in our classification. [ABSTRACT FROM AUTHOR]

Abstract For an s -tuple A = (A 1 , ... , A s) of square matrices of the same size, the (joint) determinant of A and the characteristic polynomial of A are defined by det ⁡ (A) (z) = det ⁡ (z 1 A 1 + z 2 A 2 + ⋯ + z s A s) and p A (z) = det ⁡ (z 0 I + z 1 A 1 + z 2 A 2 + ⋯ + z s A s) , respectively. This paper calculates determinant of the finite dimensional irreducible representations of sl (2 , F) , which is either zero or a product of some irreducible quadratic polynomials. Moreover, it shows that a finite dimensional Lie algebra is solvable if and only if the characteristic polynomial is completely reducible. [ABSTRACT FROM AUTHOR]

Let K be a field of characteristic zero, A = K[x1, . . ., xn] the polynomial ring and R = K(x1, .. ., xn) the field of rational functions in n variables over K. The Lie algebra Wn (K) of all K-derivations on A is of great interest since its elements may be considered as vector fields on Kn with polynomial coefficients. If L is a subalgebra of Wn (K), then one can define the rank rkA L of L over A as the dimension of the vector space RL over the field R. Finite dimensional (over K) subalgebras of Wn (K) of rank 1 over A were studied by the first author jointly with I. Arzhantsev and E. Makedonskiy. We study solvable subalgebras L of Wn (K) with rkA L = 1, without restrictions on dimension over K. Such Lie algebras are described in terms of Darboux polynomials. [ABSTRACT FROM AUTHOR]

In this paper, the maximal abelian dimension is algorithmically and computationally studied for the Lie algebra hn, of n×n upper-triangular matrices. More concretely, we define an algorithm to compute abelian subalgebras of hn besides programming its implementation with the symbolic computation package MAPLE. The algorithm returns a maximal abelian subalgebra of hn and, hence, its maximal abelian dimension. The order n of the matrices hn is the unique input needed to obtain these subalgebras. Finally, a computational study of the algorithm is presented and we explain and comment some suggestions and comments related to how it works.

This is a short note on generalized G 2 -structures obtained as a consequence of a T -dual construction given in del Barco et al. (2017). Given classical G 2 -structure on certain seven dimensional manifolds, either closed or co-closed, we obtain integrable generalized G 2 -structures which are no longer a usual one, and with non-zero three form in general. In particular we obtain manifolds admitting closed generalized G 2 -structures not admitting closed (usual) G 2 -structures. [ABSTRACT FROM AUTHOR]

Let G be a type 1 connected and simply connected solvable Lie group. The generalized moment map for π in G ^ {\widehat{G}} , the unitary dual of G, sends smooth vectors of the representation space of π to 𝒰 ⁢ (𝔤) * {{\mathcal{U}(\mathfrak{g})}^{*}} , the dual vector space of 𝒰 ⁢ (𝔤) {\mathcal{U}(\mathfrak{g})}. The convex hull of the image of the generalized moment map for π is called its generalized moment set, denoted by J ⁢ (π) {J(\pi)}. We say that G ^ {\widehat{G}} is generalized moment separable when the generalized moment sets differ for any pair of distinct irreducible unitary representations. Our main result in this paper provides a second proof of the generalized moment separability theorem for G. [ABSTRACT FROM AUTHOR]

Let N denote the class of nilpotent Lie algebras. For any finite-dimensional Lie algebra L over an arbitrary field F, there exists a smallest ideal I of L such that L/I ∈ N. This uniquely determined ideal of L is called the nilpotent residual of L and is denoted by LN. In this paper, we define the subalgebra S(L) = ∩H≤LIL(HN). Set S0(L) = 0. Define Si+1(L)/Si(L) = S(L/Si(L)) for i > 1. By S∞(L) denote the terminal term of the ascending series. It is proved that L = S∞(L) if and only if LN is nilpotent. In addition, we investigate the basic properties of a Lie algebra L with S(L) = L. [ABSTRACT FROM AUTHOR]

We investigate (co)homological properties of Lie algebras that are constructed from a finite poset: the solvable class gl≺ and the nilpotent class gl≺. We confirm the conjecture [8, 1.16(1), p. 141] that says: every prime power pr ≤ n - 2 appears as torsion in H*(niln; Z), and every prime power pr ≤ n - 1 appears as torsion in H*(soln; Z). If ≺ is a bounded poset, then the (co)homology of gl ≺ is torsion-convex, i.e., if it contains ptorsion, then it also contains p'-torsion for every prime p' < p. We obtain new explicit formulas for the (co)homology of some families over arbitrary fields. Among them are the solvable non-nilpotent analogs of the Heisenberg Lie algebras from [2], the 2-step Lie algebras from [1], strictly block-triangular Lie algebras, etc. The combinatorics of how the resulting generating functions are obtained are interesting in their own right. All this is done by using AMT (algebraic Morse theory [9, 12, 8]). This article serves as a source of examples of how to construct useful acyclic matchings, each of which in turn induces compelling combinatorial problems and solutions. It also enables graph theory to be used in homological algebra. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The fundamental theorem of cancellation AMT [4] and [11] , which is the algebraic generalization of discrete Morse theory [2] for simplicial complexes and smooth Morse theory [10] for differentiable manifolds, is discussed in the context of general chain complexes of free modules. The Chevalley (co)homology table of a Lie algebra is often a tremendous beast. Using AMT, we compute the homology of the Lie algebra of all triangular matrices sol n over Q or Z p for large enough primes p . We determine the column and row in the table of H k ( sol n ; Z ) where the p -torsion first appears. Module H k ( sol n ; Z p ) is expressed by the homology of a chain subcomplex for the Lie algebra of all strictly triangular matrices nil n , using the Künneth formula. All conclusions are accompanied by computer experiments. Then we generalize some results to Lie algebras of (strictly) triangular matrices gl n ≺ and gl n ⪯ with respect to any partial ordering ⪯ on [ n ] . We determine the multiplicative structure of H ⁎ ( gl n ⪯ ) w.r.t. the cup product over fields of zero or sufficiently large characteristic, the result being the exterior algebra. Matchings used here can be analogously defined for other Lie algebra families and in other (co)homology theories; we collectively call them normalization matchings . They are useful for theoretical as well as computational purposes. [ABSTRACT FROM AUTHOR]

A pair of sequences of nilpotent Lie algebras denoted byNn, 7andNn, 16are introduced. Here,ndenotes the dimension of the algebras that are defined forn ≥ 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 thatNn, 7andNn, 16serve 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. [ABSTRACT FROM PUBLISHER]

We give necessary and sufficient conditions of the existence of a left-invariant metric of strictly negative Ricci curvature on a solvable Lie group the nilradical of whose Lie algebra is a filiform Lie algebra . It turns out that such a metric always exists, except for in the two cases, when is one of the algebras of rank two, or , and is a one-dimensional extension of , in which cases the conditions are given in terms of certain linear inequalities for the eigenvalues of the extension derivation. [ABSTRACT FROM AUTHOR]

We study solvable Lie groups which admit a left-invariant metric of strictly negative Ricci curvature. We obtain necessary and sufficient conditions of the existence of such a metric for Lie groups the nilradical of whose Lie algebra is either abelian or Heisenberg or standard filiform and discuss some open questions. [ABSTRACT FROM AUTHOR]