Lie algebras with -action are studied.
problem Lie algebra structures on symplectic Lie groups.
method Definition of -quasi-Frobenius Lie algebras.
result Induced -action on -quasi-Frobenius Lie algebras.
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48 results for “quasi-Frobenius Lie algebras”
Identifies groupoid analogue of symplectic Lie groups.
problem Understanding symplectic structures on Lie groupoids.
method Introduces -symplectic Lie groupoids and explores their properties.
result Establishes a correspondence between Lie algebroid structures and -symplectic Lie groupoid structures.
We prove that the description of pencils of compatible (N x N)-metrics of constant Riemannian curvature is equivalent to a special class of integrable N-parametric deformations of quasi-Frobenius (in general, noncommutative) algebras.
We study -ary commutative superalgebras and -algebras that possess a skew-symmetric invariant form, using the derived bracket formalism. This class of superalgebras includes for instance Lie algebras and their -ary generalizations, commutative associative and Jordan algebras with an invariant form. We…
We investigate the properties of principal elements of Frobenius Lie algebras, following the work of M. Gerstenhaber and A. Giaquinto. We prove that any Lie algebra with a left symmetric algebra structure can be embedded, in a natural way, as a subalgebra of some sl(m,K), for K= R or C. Hence, the work of Belavin and D…
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 -type Lie algebras.
problem Investigating prolongations of Lie algebras associated with pseudo -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.
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…
A Lie-admissible algebra gives by anticommutativity a Lie algebra. In this work we study remarkable classes of Lie-admissible algebras such as Vinberg, PreLie algebras. We compute the corresponding binary quadratic operads and study their Koszul duality. Considering Lie algebras as Lie-admissible algebras we can define…
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 -type Lie algebras and their representations through Clifford algebras.
result Different types of Lie algebras have varying possibilities of containing pseudo -type Lie algebras in their negative part.
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.
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.
Study on pre-Lie structures for semisimple Lie algebras over C.
problem Admissibility of pre-Lie structures in semisimple Lie algebras.
method Examined properties of anti-flexible algebras (AFAs), computed Lie-admissibility criteria, and provided examples.
result Explicit counterexample of an AFA admissible by sl(2, C).
We introduce post-Lie algebra structures on pairs of Lie algebras $(\Lg,\Ln)$ defined on a fixed vector space . 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…
The aim of this note is to introduce the notion of a -Lie algebra and to prove some elementary properties of -Lie algebras, the category of -Lie algebras, the category of modules on a -Lie algebra and extensions of -Lie algebras. …
We construct the Lie algebra of an n-Lie algebra and we also define the notion of cohomology of an n-Lie algebra.
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…
Introduces Hom-Lie groups and their integrability, defining Hexp map and adjoint representation.
problem Integrability of Hom-Lie algebras and associated Hom-Lie groups.
method Definition of Hom-Lie groups and algebras, integration of Hom-Lie algebras, Hexp map definition.
result Every regular Hom-Lie algebra is integrable, Hexp map is universal.
Lie algebroids and curved Lie algebras are equivalent categories.
problem Understanding the relationship between Lie algebroids and curved Lie algebras.
method Developed a method to study the -category of curved Lie algebras using homotopy theory of algebras over a complete operad.
result Equivalence of -categories between Lie algebroids and certain kinds of curved Lie algebras.
The study classifies Lie algebras of vector fields in 3D.
problem Classifying Lie algebras of vector fields in 3D.
method Lifting Lie algebras from to and computing all types of transitive lifts.
result Computed all types of transitive lifts for Lie algebras from Lie's classification.
A new category of Lie algebras, called generalized Lie algebras, is presented such that classical Lie algebras and Lie-Rinehart algebras are objects of this new category. A new philosophy over generalized Lie algebroids theory is presented using the notion of generalized Lie algebra and examples of objects of the categ…
Symmetric spaces' connections form Lie admissible triple algebras.
problem Understanding the algebraic structure of symmetric spaces' connections.
method Analyzing the connection as a binary operator on tangent bundle sections, identifying Lie admissibility constraints.
result Connection algebra of symmetric spaces is a Lie admissible triple algebra.
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 and .
Lie groups of automorphisms of cotangent bundles of Lie groups are completely characterized and interesting results are obtained. We give prominence to the fact that the Lie groups of automorphisms of cotangent bundles of Lie groups are super symmetric Lie groups. In the cases of orthogonal Lie lgebras, semi-simple Lie…
Factorizable Lie bialgebras, quadratic Rota-Baxter Lie algebras and Rota-Baxter Lie bialgebrasmath-ph
The paper connects Lie bialgebras, Rota-Baxter Lie algebras, and their properties.
problem Exploring connections between Lie bialgebras and Rota-Baxter Lie algebras.
method Introducing quadratic Rota-Baxter Lie algebras, matched pairs, bialgebras, and Manin triples.
result Established a correspondence between factorizable Lie bialgebras and quadratic Rota-Baxter Lie algebras.
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 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.
Study on generalized derivations in polynomial vector fields Lie algebras.
problem Understanding generalized derivations in specific Lie sub-algebras of polynomial vector fields.
method Analysis of Lie sub-algebras containing constant and Euler vector fields, under specified conditions.
result Characterization of generalized derivations in the studied Lie sub-algebras.
Born Lie algebrasmath.DG
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.
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 can be obtained by the double extension process from flat Lorentzian nilpotent Lie algebras.
Study cohomology of hemistrict Lie 2-algebras, proving isomorphic results.
problem Understanding cohomology of hemistrict Lie 2-algebras.
method Functorial construction and isomorphism proof of cohomology.
result Cohomology of hemistrict Lie 2-algebras is isomorphic to Chevalley-Eilenberg cohomology.
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.
Abstract: Connections between Lie algebras and symplectic nilmanifolds explored.
problem Understanding the relationship between Lie algebras and symplectic nilmanifolds.
method Central extensions of Lie algebras, dynamical systems, and symplectic nilmanifolds constructed.
result Covering Lie groups for symplectic nilmanifolds can have any rank as solvable Lie groups.
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.
This research introduces Lie brackets on spaces of biderivations in Lie algebras.
problem Understanding higher-order infinitesimal symmetries in Lie algebras.
method Study of right biderivations and Lie brackets on their spaces.
result New Lie algebra framework for biderivations with applications in deformation theory.
The paper investigates gradings of complex simple Lie algebras, focusing on -gradings and their algebraic structures.
problem Investigating the algebraic structure of -gradings of complex simple Lie algebras.
method Completely determining the possible reductive algebras 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 -grading is the usual -grading of the exceptional Lie algebra .
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.
Local description of solvable Lie algebras of vector fields.
problem Understanding solvable Lie algebras of vector fields.
method Local and constructive differential geometric description.
result Implication of Lie's conjecture for solvable Lie algebras.
Study on pseudo-Riemannian algebraic Ricci solitons in 4D Lie groups.
problem Investigating conditions for pseudo-Riemannian algebraic Ricci solitons on 4D Lie algebras.
method Analyzing the algebraic Ricci soliton equation for each 4D Lie algebra.
result Complete description of pseudo-Riemannian algebraic Ricci solitons in dimension four.
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-…
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.
The paper classifies Lie algebras with special operators.
problem Classifying 3D Lie algebras with regular semisimple algebraic Nijenhuis operators.
method Described all Nijenhuis eigenbases for each 3D Lie algebra.
result Different answers in real and complex cases, some Lie algebras admit operators, others do not.
Nice bases for Lie algebrasmath.DG
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.
We obtain a characterization of the real Lie algebras admitting abelian complex structures in terms of certain affine Lie algebras , where is a commutative algebra. These affine Lie algebras are natural generalizations of and the corresponding Lie grou…
We extend the classical characterization of a finite-dimensional Lie algebra g in terms of its Maurer-Cartan algebra-the familiar differential graded algebra of alternating forms on g with values in the ground field, endowed with the standard Lie algebra cohomology operator-to sh Lie-Rinehart algebras. To this end, we …
New representations of Lie algebras via monoidal category actions.
problem Constructing representations of Lie algebras using monoidal categories.
method Using crossed homomorphisms and monoidal categories to generate representations.
result Established new bifunctor for weak and admissible representations of Lie-Rinehart algebras.