Study topological properties of integrable case on Lie algebra so(4).
problem Topological analysis of integrable case for Euler's equations on so(4).
method Construction of bifurcation diagrams, determination of critical points, description of Liouville tori bifurcations, computation of loop molecules.
result Some topological properties of Kovalevskaya case can be derived from the case on so(4).
The paper explores properties of Lie algebra g2 and related geometric structures.
problem Understanding the structure of Lie algebra g2 and its subalgebras.
method Analyzes properties of g2, constructs subalgebras, and proves canonical forms.
result An element of g2 cannot have rank 2, and if it has rank 4, its kernel is an associative subspace.
Study the topology of energy surfaces in a specific mathematical case.
problem Determine the topology of isoenergy surfaces in the Kovalevskaya integrable case on Lie algebra so(4).
method Use Fomenko-Zieschang invariants of Liouville foliations.
result Classify diffeomorphisms of three-dimensional regular level surfaces.
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.
We characterize H-like Lie algebras in terms of subspaces of cones over conjugacy classes in so(Rq), translating the classification problem for H-like Lie algebras to an equivalent problem in linear algebra. We study properties of H-like Lie algebras, present new methods for constructing them, in…
Introducing Nijenhuis forms on Lie-infinity algebras gives a general frame to understand deformations of the latter. We give here a Nijenhuis interpretation of a deformation of an arbitrary Lie algebroid into a Lie-infinity algebra. Then we show that Nijenhuis forms on Lie-infinity algebras also give a short and effici…
Let g be a simplicial Lie algebra with Moore complex Ng of length k. Let G be the simplicial Lie group integrating g, which is simply connected in each simplicial level. We use the 1-jet of the classifying space of G to construct, starting from g, a Lie k-algebra L. The so constructed Lie k-algebra L is actually a diff…
The present paper contains a systematic study of the structure of metric Lie algebras, i.e., finite-dimensional real Lie algebras equipped with a non-degenerate invariant symmetric bilinear form. We show that any metric Lie algebra without simple ideals has the structure of a so called balanced quadratic extension of a…
This overview paper is intended as a quick introduction to Lie algebras of vector fields. Originally introduced in the late 19th century by Sophus Lie to capture symmetries of ordinary differential equations, these algebras, or infinitesimal groups, are a recurring theme in 20th-century research on Lie algebras. I will…
A class of Z_2-graded Lie algebra and Lie superalgebra extensions of the pseudo-orthogonal algebra of a spacetime of arbitrary dimension and signature is investigated. They have the form g = g_0 + g_1, with g_0 = so(V) + W_0 and g_1 = W_1, where the algebra of generalized translations W = W_0 + W_1 is the maximal solva…
Triality connects three polynomial bases in Lie algebra studies.
problem Understanding relationships between invariant polynomials.
method Triality of mSO(8,C) and Hitchin fibrations. result Explicit automorphisms linking polynomial bases identified.
We propose a new definition of so called Hamiltonian forms in n-plectic geometry and show that they have a non-trivial Lie infinity-algebra structure.
We show how to integrate a weak morphism of Lie algebra crossed-modules to a weak morphism of Lie 2-groups. To do so we develop a theory of butterflies for 2-term L_infty algebras. In particular, we obtain a new description of the bicategory of 2-term L_infty algebras. We use butterflies to give a functorial constructi…
In this paper we aim to understand the category of stable-Yetter-Drinfeld modules over enveloping algebra of Lie algebras. To do so, we need to define such modules over Lie algebras. These two categories are shown to be isomorphic. A mixed complex is defined for a given Lie algebra and a stable-Yetter-Drinfeld module o…
The study explores Lie superalgebras constructed from Lie algebras using Schouten-like brackets.
problem Investigate how the core Lie algebra controls the Lie superalgebra.
method Construct Lie superalgebras from abstract Lie algebras using Schouten-like brackets and analyze Betti numbers of super homology groups.
result For low dimensional non-abelian Lie algebras, the Betti numbers of super homology groups provide insights into the control of the core Lie algebra.
New examples of Lie algebras with ad-invariant metrics found.
problem Finding ad-invariant metrics on nonnice nilpotent Lie algebras.
method Introducing single extension method to construct Lie algebras with ad-invariant metrics.
result Explicit examples of nonnice nilpotent Lie algebras with ad-invariant metrics for dimensions > 10 and steps > 2.
Classifies Lie bialgebras using Darboux families.
problem Classifying real four-dimensional indecomposable coboundary Lie bialgebras.
method Introducing Darboux families to classify Lie bialgebras geometrically.
result Classification of coboundary Lie bialgebras on real four-dimensional indecomposable Lie algebras.
If a Lie algebra structure g on a vector space is the sum of a family of mutually compatible Lie algebra structures g_i's, we say that g is simply assembled from the g_i's. Repeating this procedure with a number of Lie algebras, themselves simply assembled from the g_i's, one obtains a Lie algebra assembled in two step…
The branching problem for a couple of non-compatible Lie algebras and their parabolic subalgebras applied to generalized Verma modules was recently discussed in \cite{ms}. In the present article, we employ the recently developed F-method, \cite{KOSS1}, \cite{KOSS2} to the couple of non-compatible Lie algebras $({\LieGt…
Study on deformations of symmetric spaces using Jordan algebras.
problem Deformability of symmetric Einstein metrics on compact Lie algebras.
method Developed sandwich operators and quadratic Casimir operators for compact Lie algebras; calculated obstruction integrals from invariant polynomials; explored relation to simple Jordan algebras.
result Proved the nonlinear instability of most infinitesimally deformable irreducible compact symmetric spaces.
Let G be a Lie group. On the trivial principal G-bundle over the Lie algebra of G there is a natural connection whose curvature is the Lie bracket. The exponential map is given by parallel transport of this connection. If G is the diffeomorphism group of a manifold, the curvature of the natural connection is the Lie br…
In this paper we deal with symplectic Lie algebras. All symplectic structures are determined for dimension four and the corresponding Lie algebras are classified up to equivalence. Symplectic four dimensional Lie algebras are described either as solutions of the cotangent extension problem or as symplectic double exten…
We construct many examples of Lie groups with compact Levi factor admitting a left-invariant metric with negative Ricci curvature. We start with a Lie algebra with Levi factor su(n) or so(n) acting on an abelian nilradical via the representation on the space of homogeneous polynomials. In the case of su(2) we obtain a …
The paper integrates Rota-Baxter Lie algebras into Lie group structures and geometries.
problem Integrating Rota-Baxter operators into Lie group structures and geometries.
method Introducing Rota-Baxter operators on Lie groups, Lie algebroids, and groupoids.
result Geometrization of Rota-Baxter Lie algebras and groups.
The Kontsevich integral of a knot is a powerful invariant which takes values in an algebra of trivalent graphs with legs. Given a Lie algebra, the Kontsevich integral determines an invariant of knots (the so-called colored Jones function) with values in the symmetric algebra of the Lie algebra. Recently A. Kricker and …
We show that the canonical central extension of the group of sections of a Lie group bundle over a compact manifold, constructed in [NW09], is universal. In doing so, we prove universality of the corresponding central extension of Lie algebras in a slightly more general setting.
We define an n-plectic structure as a commutative and torsionless Lie Rinehart pair, together with a distinguished cocycle from its Chevalley-Eilenberg complex. This 'n-plectic cocycle' gives rise to an extension of the Chevalley-Eilenberg complex by so called symplectic tensors. The cohomology of this extension genera…
We provide an explicit algorithm to calculate invariant tensors for the adjoint representation of the simple Lie algebra sl(n), as well as arbitrary representation in terms of roots. We also obtain explicit formulae for the adjoint representations of the orthogonal and symplectic Lie algebras so(n) and sp(n).
A Lie system is the non-autonomous system of differential equations describing the integral curves of a non-autonomous vector field taking values in a finite-dimensional Lie algebra of vector fields, a so-called Vessiot--Guldberg Lie algebra. This work pioneers the analysis of Lie systems admitting a Vessiot--Guldberg …
Higher homotopy generalizations of Lie-Rinehart algebras, Gerstenhaber-, and Batalin-Vilkovisky algebras are explored. These are defined in terms of various antisymmetric bilinear operations satisfying weakened versions of the Jacobi identity, as well as in terms of operations involving more than two variables of the L…
This work merges 3-anchored bundles into 3-Lie algebroids.
problem Combining two 3-anchored bundles into a unified structure.
method Develops algebraic framework for merging bundles with mutual actions and cocycle terms.
result Unified setting for 3-Lie algebroids and special cases.
Study finds 132 complex invariant Einstein metrics on a specific flag manifold and constructs Ricci-flat metrics.
problem Identifying complex invariant Einstein metrics on a specific flag manifold.
method Inonu-Wigner contractions of Lie algebras.
result 132 complex invariant Einstein metrics found on the manifold.
This paper studies actions of solvable Lie groups on nilpotent Lie groups.
problem Characterizing which solvable Lie groups can act simply transitively on nilpotent Lie groups.
method Using Lie algebra properties and semisimple splitting, the paper provides methods to check for such actions.
result A full description of possibilities for actions up to dimension 4.
Unified approach to deform Lie-Hamilton systems using Poisson-Hopf algebra.
problem Deforming Lie systems with quantum algebras.
method Poisson-Hopf algebra deformations applied to Lie-Hamilton systems.
result Unified approach to deformations of Lie-Hamilton systems on the real plane.
Vogel's construction links knot invariants to Lie algebras, revealing new insights.
problem Can all finite type knot invariants be derived from Lie algebras?
method Parameterized expansion coefficients with three parameters and constructed a polynomial to vanish for all simple Lie algebras.
result Vogel's construction implies an alternative axiomatization of simple Lie algebras.
Let M be a compact simply connected hyperkähler (or holomorphically symplectic) manifold, \dim H^2(M)=n. Assume that M is not a product of hyperkaehler manifolds. We prove that the Lie algebra so(n-3,3) acts by automorphisms on the cohomology ring H^*(M). Under this action, the space H^2(M) is isomorphic to the fundame…
A Lie system is a system of first-order ordinary differential equations describing the integral curves of a t-dependent vector field taking values in a finite-dimensional real Lie algebra of vector fields: a so-called Vessiot-Guldberg Lie algebra. We suggest the definition of a particular class of Lie systems, the $k…
New examples of rigid Lie foliations with dense leaves found.
problem Infinitesimal rigidity of Lie foliations with dense leaves.
method Construction of specific Lie foliations.
result First examples of infinitesimally rigid Riemannian foliations with dense leaves.
The cohomology of affine structures on Lie groups is compared with that of Koszul-Vinberg algebras.
problem Comparing De Rham cohomology and KV-cohomology on Lie groups SO(2), H3(R), and Galilei group SGal(3).
method Constructing a three vertex directed graph connecting associative algebras, KV-cohomology, and Lie groups.
result Evaluating the algebraic quotient of the cohomology groups.
Constructing transitive nilpotent Lie algebras from dilations and analyzing their prolongations.
problem Understanding the derivations of Tanaka prolongations of transitive nilpotent Lie algebras.
method Constructing transitive nilpotent Lie algebras from dilations and analyzing their prolongations.
result Derivations of degree 0 are given by vector fields of degree 0, and the Tanaka prolongation recovers the whole algebra of polynomial vectors defined by the dilation.
In the present paper we study the rigidity of 2-step Carnot groups, or equivalently, of graded 2-step nilpotent Lie algebras. We prove the alternative that depending on bi-dimensions of the algebra, the Lie algebra structure makes it either always of infinite type or generically rigid, and we specify the bi-dimensions …
Study on symmetries and equilibria in Poisson manifolds, with applications to rigid body dynamics.
problem Characterizing conformal relative equilibria on Poisson manifolds.
method Introducing conformally Poisson actions and momentum maps, establishing algebraic criteria.
result Classification of nontrivial conformal relative equilibria in Lie algebras, with applications to rigid body dynamics.
Infinitesimal conformal transformations of Rn are always polynomial and finitely generated when n>2. Here we prove that the Lie algebra of infinitesimal conformal polynomial transformations over Rn, n>1, is maximal in the Lie algebra of polynomial vector fields. When n is greater than 2 and p,q are such t…
We characterise the canonical elements, in the sense of Burstall--Rawnsley \cite{BurRaw90}, of a compact semisimple Lie algebra and discuss the case of so(n) in detail. In so doing, we correct two errors in Burstall et al. \cite{BurEscFerTri04}.
Study on deformation of affine structures on Lie groups using cohomology.
problem Understanding the gap between global topological invariants and left-invariant affine structures.
method Comparing De Rham and Koszul-Vinberg cohomology groups on Lie groups SO(2), H3(R), and SGal(3).
result Vanishing theorem for the second KV-cohomology group, proving structural rigidity of orbits.
This paper shows how post-Lie algebra structures can be induced by simply transitive NIL-affine actions.
problem Understanding which solvable Lie groups can act simply transitively on nilpotent Lie groups.
method Introducing post-Lie algebra structures and showing their correspondence with simply transitive actions.
result Simply transitive NIL-affine actions induce complete post-Lie algebra structures in the 2-step nilpotent case.
Abstract: Tangent categories get a Cartan calculus with scalar multiplication by a commutative ring.
problem Constructing a Cartan calculus in tangent categories.
method Define scalar multiplication by a commutative ring object R to equip tangent bundles with R-module structure. result Every object in tangent categories carries a Cartan calculus of Lie-Rinehart forms.
The preprint is prepared as description of results that were obtained during joint scientific project No: 71NC /2015/VNCCCT on the VIASM (Vietnam Institute for Advanced Study in Mathematics) from 08.12.2015 to 06.02.2016. The problem was formulated how to calculate so called the Mackenzie obstruction for existing of tr…