Characterizes Lie groups with specific structures and finds a correspondence between carrollian and galilean Lie algebras.
problem Understanding Lie groups with specific structures.
method Using structure theory of metric Lie algebras and defining new Lie algebras with skew-symmetric derivations.
result A canonical correspondence between carrollian and galilean Lie algebras mediated by bargmannian Lie algebras.
The paper studies prolongations of Lie algebras associated with pseudo H-type Lie algebras.
problem Investigating prolongations of Lie algebras associated with pseudo H-type Lie algebras. method Analyzing prolongations of associated fundamental graded Lie algebra and associated conformal pseudo-subriemannian fundamental graded Lie algebra.
result The prolongation of the associated conformal pseudo-subriemannian fundamental graded Lie algebra coincides with that of the associated fundamental graded Lie algebra under certain conditions.
Develops a bialgebra theory for post-Lie algebras using geometric interpretations and bilinear forms.
problem Characterizing and understanding post-Lie algebras and their associated structures.
method Utilizes Manin triples and generalized Hessian Lie groups to define and characterize post-Lie algebras with nondegenerate symmetric invariant bilinear forms.
result Establishes a bialgebra theory for post-Lie algebras via the Manin triple approach, including new algebraic structures like pp-post-Lie algebras.
The paper generalizes para-Kähler Lie algebras to k-para-Kähler Lie algebras and explores their structures.
problem Characterizing and understanding k-para-Kähler Lie algebras.
method Generalization of para-Kähler Lie algebras to k-para-Kähler Lie algebras, introduction of new structures, determination of Lie algebras.
result Determination of all k-symplectic Lie algebras of dimension (k+1) and six-dimensional 2-para-Kähler Lie algebras.
New theory disassembles Lie algebras into dyons and triadons.
problem Understanding the structure of Lie algebras through disassembly.
method Modular disassembly of Lie algebras into dyons and triadons.
result Any Lie algebra can be assembled from dyons and triadons.
We construct the Lie algebra of an n-Lie algebra and we also define the notion of cohomology of an n-Lie algebra.
Study on Lie algebras from Clifford modules, focusing on specific types.
problem Characterizing Lie algebras from Clifford modules and their graded structures.
method Analysis of pseudo H-type Lie algebras and their representations through Clifford algebras. result Different types of Lie algebras have varying possibilities of containing pseudo H-type Lie algebras in their negative part. New Lie algebras from knot homology.
problem Defining Lie algebras from knot homology.
method Using group homology, analogous to Goldman Lie algebra.
result Relations among new Lie algebras discussed.
Proofs centerless unimodular contact Lie algebras.
problem Characterizing centerless unimodular contact Lie algebras.
method Elementary proof and introduction of DS-contact Lie algebras.
result The only centerless unimodular examples are sl(2,R) and su(2). Classifies symplectic Lie algebras with degenerate center.
problem Understanding symplectic Lie algebras with degenerate center.
method Standard model, quadratic cohomology sets, classification scheme.
result Complete list of 6-dimensional nilpotent symplectic Lie algebras.
Study coKähler structures on Lie algebras using Fino-Vezzoni correspondence.
problem Characterize coKähler structures on Lie algebras.
method Use Fino-Vezzoni correspondence to relate coKähler Lie algebras to Kähler Lie algebras.
result Complete the flat case for odd-dimensional Lie algebras, proving coKähler structures exist.
Born Lie algebras classified up to 6D, with integrable metrics studied.
problem Classifying and understanding Born Lie algebras.
method Bicross product construction from pseudo-Riemannian Lie algebras.
result Classification of Lie algebras up to 6D with integrable Born structures.
In this paper, we present a study on the prolongations of representations of Lie algebras. We show that a tangent bundle of a given Lie algebra attains a Lie algebra structure. Then, we prove that this tangent bundle is algebraically isomorphic to the Lie algebra of a tangent bundle of a Lie group. Using these, we defi…
We introduce post-Lie algebra structures on pairs of Lie algebras $(\Lg,\Ln)$ defined on a fixed vector space V. Special cases are LR-structures and pre-Lie algebra structures on Lie algebras. We show that post-Lie algebra structures naturally arise in the study of NIL-affine actions on nilpotent Lie groups. We obtai…
Uniform Lie algebras and graphs linked via combinatorial methods.
problem Defining and classifying uniform Lie algebras.
method Combining combinatorial definitions and graph theory.
result Established correspondence between uniform Lie algebras and uniformly colored graphs.
Generalizes Lie algebra of conformal Killing vector fields to conformal Killing-Yano forms.
problem Understanding the algebraic structure of conformal Killing-Yano forms.
method Proposes a new Lie bracket and shows graded Lie algebra properties in constant and Einstein manifolds.
result Conformal Killing-Yano forms form a graded Lie algebra in specific manifolds.
The study classifies Lie algebras of vector fields in 3D.
problem Classifying Lie algebras of vector fields in 3D.
method Lifting Lie algebras from C2 to C2imesC and computing all types of transitive lifts. result Computed all types of transitive lifts for Lie algebras from Lie's classification.
We study post-Lie algebra structures on pairs of Lie algebras (g,n), and prove existence results for the case that one of the Lie algebras is semisimple. For semisimple g and solvable n we show that there exist no post-Lie algebra structures on (g,n). For semisimple n and certain solvable g we construct canonical post-…
Classifies metric symplectic Lie algebras with quadratic extensions.
problem Classifying metric symplectic Lie algebras.
method Standard model, quadratic cohomology sets, isomorphism classes.
result Complete list of metric symplectic Lie algebras in special cases.
The study characterizes flat pseudo-Euclidean Lie algebras and their properties.
problem Characterizing flat pseudo-Euclidean Lie algebras and their properties.
method Using the double extension process and analyzing the center of the Lie algebras.
result All flat pseudo-Euclidean nilpotent Lie algebras of signature (2,n−2) can be obtained by the double extension process from flat Lorentzian nilpotent Lie algebras. Study locally conformally balanced metrics on specific Lie algebras.
problem Characterize and classify locally conformally balanced metrics on almost abelian Lie algebras.
method Characterizations and classifications based on specific properties of Lie algebras.
result Classification of six-dimensional almost abelian Lie algebras with locally conformally balanced metrics.
The paper studies the center of the Goldman Lie algebra and its properties.
problem Identifying the center of the Goldman Lie algebra and its properties.
method Analyzing the Goldman Lie algebra as a Z_2-graded Lie algebra and using properties of the even part.
result The center of the even part of the Goldman Lie algebra is generated by specific classes of loops.
We present structural properties of Lie algebras admitting symmetric, invariant and nondegenerate bilinear forms. We show that these properties are not satisfied by nilradicals of parabolic subalgebras of real split forms of complex simple Lie algebras, neither by 2-step nilpotent Lie algebras associated with graphs, w…
Study SKT and Kähler structures on specific Lie algebras.
problem Characterize SKT and Kähler structures on solvable Lie algebras with codimension two nilradical.
method Classify and construct new examples of SKT solvable Lie algebras.
result Provide a classification of SKT Lie algebras in dimension six and extend SKT nilpotent Lie algebras to higher dimensions.
The notion of Poisson manifold with compatible pseudo-metric was introduced by the author in [1]. In this paper, we introduce a new class of Lie algebras which we call a pseudo-Rieamannian Lie algebras. The two notions are strongly related: we prove that a linear Poisson structure on the dual of a Lie algebra has a com…
The aim of this note is to introduce the notion of a D-Lie algebra and to prove some elementary properties of D-Lie algebras, the category of D-Lie algebras, the category of modules on a D-Lie algebra and extensions of D-Lie algebras. …
The paper studies nice bases for Lie algebras and their properties.
problem Existence and number of nice bases on Lie algebras.
method Examined three classes of Lie algebras: direct sums, almost abelian ones, and those associated to graphs.
result For every natural number n, an indecomposable Lie algebra exists with exactly n nice bases.
Defines and analyzes the holonomy Lie algebra of geometric lattices.
problem Holonomy Lie algebra of geometric lattices.
method Combinatorial definition and analysis of solvable pairs of lattices.
result Holonomy Lie algebra structure of hypersolvable lattices.
H-type Lie algebras were introduced by Kaplan as a class of real Lie algebras generalizing the familiar Heisenberg Lie algebra h3. The H-type property depends on a choice of inner product on the Lie algebra g. Among the H-type Lie algebras are the complex Heisenberg Lie algebras $\mathfrak{h}…
Characterizes G2-structures on Lie algebras with non-trivial center.
problem Classifying Lie algebras with G2-structures.
method Analyzing Lie algebras with non-trivial center, using contactization and symplectic properties.
result Six unimodular Lie algebras with non-trivial center admit closed G2-structures.
Study on symplectic Lie algebras with specific dimensions.
problem Understanding symplectic structures on solvable Lie algebras.
method Analysis of Lie algebras over R or C with specific properties. result Description of complete Lie algebras with dim nilradical ≤ 6 and symplectic structure.
If a Lie algebra structures $\gG$ on a vector space is the sum of a family of mutually compatible Lie algebra structures $\gG_i$, we say that $\gG$ is \emph{simply assembled} from $\gG_s$'s. By repeating this procedure several times one gets a family of Lie algebras \emph{assembled} from $\gG_s$'s. The central result o…
The study examines extensions of Lie algebras with specific geometric structures.
problem Conditions for preserving geometric structures in Lie algebra extensions.
method Analyzes extensions of Sasakian and Frobenius-Kähler Lie algebras.
result Conditions for maintaining Sasakian or Frobenius-Kähler structures after extensions.
2-compatible Lie algebras are quadratic deformations of Lie algebras with specific constraints.
problem Classifying contact Lie algebras using quadratic deformations.
method Defining 2-compatible Lie algebras as quadratic deformations of Lie algebras and studying the constraints on these deformations.
result Any (2p+1)-dimensional contact Lie algebra is isomorphic to a quadratic deformation of the Heisenberg algebra.
New abelian quotient found in symplectic derivation Lie algebra.
problem Understanding the structure of symplectic derivation Lie algebra.
method Computational approach to abelianization of the weight 12 part.
result 1-dimensional weight 12 part for g≥8. Method constructs complex symplectic Lie algebras from simpler ones.
problem Classifying complex symplectic Lie algebras of various dimensions.
method Complex symplectic oxidation method
result Classification of eight-dimensional nilpotent complex symplectic Lie algebras.
The paper constructs new algebraic structures from Lie algebras and ternary Nambu-Lie algebras, leading to Yang-Baxter operators.
problem Constructing new algebraic structures from Lie algebras and ternary Nambu-Lie algebras.
method Using compositions of binary Lie algebras, 3-Lie algebras, and ternary Nambu-Lie algebras, the paper constructs ternary self-distributive objects and Yang-Baxter operators.
result The constructed Yang-Baxter operators are not gauge equivalent to the transposition operator and can be deformed to new solutions.
Statistical Lie algebras with constant curvature are linked to locally conformally Kähler structures.
problem Characterizing Lie algebras with constant curvature and their geometric implications.
method Constructing statistical manifolds and Sasakian structures to relate Lie algebras to locally conformally Kähler structures.
result Statistical Lie algebras of constant curvature correspond to locally conformally Kähler Lie algebras.
Lie's Third Theorem, asserting that each finite-dimensional Lie algebra is the Lie algebra of a Lie group, fails in infinite dimensions. The modern account on this phenomenon is the integration problem for central extensions of infinite-dimensional Lie algebras, which in turn is phrased in terms of an integration proce…
Classifies special Lie algebras with semisimple types.
problem Classifying Lie algebras of semisimple type.
method Introduced conformal pseudo-subriemannian fundamental graded Lie algebras and provided their classification.
result Classification of conformal pseudo-subriemannian fundamental graded Lie algebras of semisimple type and their prolongations.
The paper investigates gradings of complex simple Lie algebras, focusing on ∣3∣-gradings and their algebraic structures.
problem Investigating the algebraic structure of ∣3∣-gradings of complex simple Lie algebras. method Completely determining the possible reductive algebras n0 and proving the uniqueness of a specific free nilpotent Lie algebra. result The only free nilpotent Lie algebra of step 3 that appears as the negative part of a ∣3∣-grading is the usual ∣3∣-grading of the exceptional Lie algebra g2. Study η-Einstein Sasakian structures on Lie algebras, dividing cases based on center dimension.
problem Investigate η-Einstein Sasakian structures on Lie algebras. method Divide cases based on center dimension, use theory of normal j-algebras and modifications of Hermitian Lie algebras to construct examples. result Construct new examples of η-Einstein Sasakian Lie algebras and solvmanifolds. Study on a specific type of Lie algebras with Kähler and contact properties.
problem Characterizing and classifying transversely Kähler almost contact metric Lie algebras.
method Analyzing properties of Lie algebras with contact forms and Kähler structures, considering center dimensions and quotient properties.
result Classification of 5-dimensional η-Einstein transversely Kähler almost contact metric Lie algebras.
The paper explores structures on 3-Lie algebras, including product, complex, and symplectic.
problem Exploring structures on 3-Lie algebras.
method Introducing phase spaces, product structures, complex structures, and compatibility conditions.
result Four types of special integrability conditions for product, complex, and symplectic structures on 3-Lie algebras.
New methods classify H-like Lie algebras with rank 2 maps.
problem Classifying H-like Lie algebras.
method Using linear algebra, studying properties, constructing with tensor products and central sums, classifying based on rank 2 maps.
result Classified H-like Lie algebras with rank 2 maps.
Complex and Kahler structures defined on hom-Lie algebras.
problem Defining structures on hom-Lie algebras.
method Introducing complex and Hermitian structures on hom-Lie algebras, providing examples, and constructing phase spaces.
result No proper complex (Hermitian) hom-Lie algebra of dimension two exists.
We study a natural Lie algebra structure on the free vector space generated by all rooted planar trees as the associated Lie algebra of the nonsymmetric operad (non-Σ operad, preoperad) of rooted planar trees. We determine whether the Lie algebra and some related Lie algebras are finitely generated or not, and prove …
Paper generalizes results from nilpotent Lie algebras to broader types.
problem Generalizing results from nilpotent Lie algebras to broader types.
method Examined Einstein Lorentzian unimodular and solvable Lie algebras.
result Key results from nilpotent Lie algebras still hold in broader settings.