arXiv research

Geometrically proves Lie algebras are identified by their Iwasawa subalgebras.

A metric Lie algebra g is a Lie algebra equipped with an inner product. A subalgebra h of a metric Lie algebra g is said to be totally geodesic if the Lie subgroup corresponding to h is a totally geodesic submanifold relative to the left-invariant Riemannian metric defined by the inner product, on the simply connected …

Establishes necessary and sufficient conditions for smooth triviality of Lie subalgebras and Lie ideals, and proves Moser's trick for foliations.

Proofs Lie's classification of certain vector field subalgebras.

The abstract discusses convergent realizations of Lie subalgebras in control theory.

Reductive (or semisimple) algebraic groups, Lie groups and Lie algebras have a rich geometry determined by their parabolic subgroups and subalgebras, which carry the structure of a building in the sense of J. Tits. We present herein an elementary approach to the geometry of parabolic subalgebras, over an arbitrary fiel…

We introduce the notion of a subregular subalgebra, which we believe is useful for classification of subalgebras of Lie algebras. We use it to construct a non-regular invariant generalized complex structure on a Lie group. As an illustration of the study of invariant generalized complex structures, we compute them all …

We determine the maximal dimension of totally geodesic subalgebras of N-graded filiform Lie algebras, and we show that these bounds are attained.

The paper analyzes symmetries of Vaidya-Bonner geodesics.

Using the basic Lie symmetry method, we find the most general Lie point symmetries group of the $\nabla u=f(u)$ Poisson's equation, which has a subalgebra isomorphic to the $3-$dimensional special Euclidean group ${\rm SE}(3)$ or group of rigid motions of ${\Bbb R}^3$. Looking the adjoint representation of ${\rm SE}(3)…

The paper explores properties of Lie algebra g2 and related geometric structures.

In this paper we describe all the nilradicals of parabolic subalgebras of split real simple Lie algebras admitting symplectic structures. The main tools used to obtain this list are Kostant's description of the highest weight vectors (hwv) of the cohomology of these nilradicals and some necessary conditions obtained fo…

The elaboration of new quantization methods has recently developed the interest in the study of subalgebras of the Lie algebra of polynomial vector fields over a Euclidean space. In this framework, these subalgebras define maximal equivariance conditions that one can impose on a linear bijection between observables tha…

Study 2D viscoelastic equations using Lie group theory.

Formula derived for Lie algebra E8's bracket.

We prove a Frobenius theorem for Banach distributions on manifolds that are modelled over locally convex spaces. Moreover, we recall how Frobenius theorems can be applied to infinite-dimensional Lie groups and obtain, that given a Lie subalgebra of the Lie algebra of a Lie group that is modelled over a locally convex s…

We introduce and study a construction of higher derived brackets generated by a (not necessarily inner) derivation of a Lie superalgebra. Higher derived brackets generated by an element of a Lie superalgebra were introduced in our earlier work. Examples of higher derived brackets naturally appear in geometry and mathem…

It is the aim of this work to study product structures on four dimensional solvable Lie algebras. We determine all possible paracomplex structures and consider the case when one of the subalgebras is an ideal. These results are applied to the case of Manin triples and complex product structures. We also analyze the thr…

Classifies 2-solvable Frobenius Lie algebras based on endomorphisms.

In the present report, by using the Stokes-Helmholtz decomposition theorem the 3-dimensional Navier-Stokes equation (NSE) is uncoupled and transformed into a scalar equation for the velocity potential when the flow field is toroidal. The dynamics of the velocity potential is independent of the vector potential. The red…

We establish the submaximal symmetry dimension for Riemannian and Lorentzian conformal structures. The proof is based on enumerating all subalgebras of orthogonal Lie algebras of sufficiently large dimension and verifying if they stabilize a non-zero Weyl tensor up to scale. Our main technical tools include Dynkin's cl…

This work is concerned with Mishchenko-Fomenko subalgebras and their restrictions to the adjoint orbits in a finite-dimensional complex semisimple Lie algebra. In this setting, it is known that each Mishchenko-Fomenko subalgebra restricts to a completely integrable system on every orbit in general position. We improve …

Constructs Lie algebras from labeled directed graphs and identifies properties of these algebras.

The purpose of this paper is to establish a connection between various subjects such as dynamical r-matrices, Lie bialgebroids, and Lagrangian subalgebras. Our method relies on the theory of Dirac structures developed in dg-ga/9508013 and dg-ga/9611001. In particular, we give a new method of classifying dynamical r-mat…

Results describing Lie ideals and maximal finite-codimensional Lie subalgebras of the Lie algebras associated with Lie algebroids with non-singular anchor maps are presented. It is also proved that every isomorphism of such Lie algebras induces a diffeomorphism of base manifolds respecting the generalized foliations de…

Proofs and descriptions of totally geodesic submanifolds in symmetric spaces.

We introduce three non-trivial 2-cocycles $c_k$, k=0,1,2, on the Lie algebra $S^3H=Map(S^3,H)$ with the aid of the corresponding basis vector fields on $S^3$, and extend them to 2-cocycles on the Lie algebra $S^3gl(n,H)=S^3H \otimes gl(n,C)$. Then we have the corresponding central extension $S^3gl(n,H)\oplus \oplus_k (…

The holonomy algebra $\g$ of an indecomposable Lorentzian (n+2)-dimensional manifold $M$ is a weakly-irreducible subalgebra of the Lorentzian algebra $\so_{1,n+1}$. L. Berard Bergery and A. Ikemakhen divided weakly-irreducible not irreducible subalgebras into 4 types and associated with each such subalgebra $\g$ a suba…

New Lie theoretic proof for complex homogeneous manifolds.

We study Poisson symmetric spaces of group type with Cartan subalgebra "adapted" to the Lie cobracket.

Study on complex curves in hypercomplex nilmanifolds with quaternionic-solvable Lie algebras.

Extended current algebra on S^3 with new bilinear form and 2-cocycle.

Study invariant Poisson structures on homogeneous manifolds, algebraically and geometrically.

We study Lagrangian subalgebras of a semisimple Lie algebra with respect to the imaginary part of the Killing form. We show that the variety $\Lagr$ of Lagrangian subalgebras carries a natural Poisson structure $Π$. We determine the irreducible components of $\Lagr$, and we show that each irreducible component is a smo…

In this paper, Lie symmetry group method is applied to find the lie point symmetries group of a PDE system that is determined general form of four-dimensional Einstein Walker manifold. Also we will construct the optimal system of one-dimensional Lie subalgebras and investigate some of its group invariant solutions.

Classifies homogeneous hypersurfaces in specific 4D geometries.

The space of vector-valued forms on any manifold is a graded Lie algebra with respect to the Frolicher-Nijenhuis bracket. In this paper we consider multiplicative vector-valued forms on Lie groupoids and show that they naturally form a graded Lie subalgebra. Along the way, we discuss various examples and different char…

We provide a quasi-Poisson version of the Drinfeld's correspondence between Poisson homogeneous spaces and Lagrangian subalgebras.

Groups with certain properties have invariant subalgebra rigidity.

We describe a class (called regular) of invariant generalized complex structures on a real semisimple Lie group G. The problem reduces to the description of admissible pairs (\gk, ω), where \gk is an appropriate regular subalgebra of the complex Lie algebra \gg^{C} associated to G and ωis a closed 2-form on \gk, such t…

A well known result of Drinfeld classifies Poisson Lie groups $(H,Π)$ in terms of Lie algebraic data in the form of Manin triples $(\mathfrak{d},\mathfrak{g},\mathfrak{h})$; he also classified compatible Poisson structures on $H$-homogeneous spaces $H/K$ in terms of Lagrangian subalgebras $\mathfrak{l}\subset\mathfrak{…

We obtain polynomial Frobenius manifolds from classical $W$-algebras associated to regular nilpotent elements in simple Lie algebras using the related opposite Cartan subalgebras.

The notions of \emph{Poisson Lie group} and \emph{Poisson homogeneous space} are extended to the Dirac category. The theorem of Drinfel$'$d (\cite{Drinfeld93}) on the one-to-one correspondence between Poisson homogeneous spaces of a Poisson Lie group and a special class of Lagrangian subalgebras of the Lie bialgebra as…

We study Lie algebras of generators of infinitesimal symmetries of almost-cosymplectic-contact structures of odd dimensional manifolds. The almost-cosymplectic-contact structure admits on the sheaf of pairs of 1-forms and functions the structure of a Lie algebra. We describe Lie subalgebras in this Lie algebra given by…

Establishing Hom-versions of Bochner theorems in pseudo-Riemannian Hom-Lie algebras

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…

Classifies non-integrable distributions with simple infinite-dimensional Lie superalgebras of symmetries.

For a perfect Lie algebra $\mathfrak{h}$ we classify all Lie algebras containing $\mathfrak{h}$ as a subalgebra of codimension $1$. The automorphism groups of such Lie algebras are fully determined as subgroups of the semidirect product $\mathfrak{h} \ltimes (k^* \times {\rm Aut}_{\rm Lie} (\mathfrak{h}))$. In the non-…