scholarly journals On the Orbits of One Non-Solvable 5-Dimensional Lie Algebra

2019 ◽  
pp. 5-20
Author(s):  
Artem Atanov ◽  
Alexander Loboda
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Fourth Order ◽  
Algebraic Surface ◽  
Vector Fields ◽  
Symmetry Algebra ◽  
Basis Vector ◽  
Invariant Polynomial ◽  
Real Hypersurfaces ◽  
Symmetry Algebras

This paper studies holomorphic homogeneous real hypersurfaces in C3 associated with the unique non-solvable indecomposable 5-dimensional Lie algebra 𝑔5 (in accordance with Mubarakzyanov’s notation). Unlike many other 5-dimensional Lie algebras with “highly symmetric” orbits, non-degenerate orbits of 𝑔5 are “simply homogeneous”, i.e. their symmetry algebras are exactly 5-dimensional. All those orbits are equivalent (up to holomorphic equivalence) to the specific indefinite algebraic surface of the fourth order. The proofs of those statements involve the method of holomorphic realizations of abstract Lie algebras. We use the approach proposed by Beloshapka and Kossovskiy, which is based on the simultaneous simplification of several basis vector fields. Three auxiliary lemmas formulated in the text let us straighten two basis vector fields of 𝑔5 and significantly simplify the third field. There is a very important assumption which is used in our considerations: we suppose that all orbits of 𝑔5 are Levi non-degenerate. Using the method of holomorphic realizations, it is easy to show that one need only consider two sets of holomorphic vector fields associated with 𝑔5. We prove that only one of these sets leads to Levi non-degenerate orbits. Considering the commutation relations of 𝑔5, we obtain a simplified basis of vector fields and a corresponding integrable system of partial differential equations. Finally, we get the equation of the orbit (unique up to holomorphic transformations) (𝑣 − 𝑥2𝑦1)2 + 𝑦2 1𝑦2 2 = 𝑦1, which is the equation of the algebraic surface of the fourth order with the indefinite Levi form. Then we analyze the obtained equation using the method of Moser normal forms. Considering the holomorphic invariant polynomial of the fourth order corresponding to our equation, we can prove (using a number of results obtained by A.V. Loboda) that the upper bound of the dimension of maximal symmetry algebra associated with the obtained orbit is equal to 6. The holomorphic invariant polynomial mentioned above differs from the known invariant polynomials of Cartan’s and Winkelmann’s types corresponding to other hypersurfaces with 6- dimensional symmetry algebras.

scholarly journals Classification of Symmetry Lie Algebras of the Canonical Geodesic Equations of Five-Dimensional Solvable Lie Algebras

Symmetry ◽  
2019 ◽  
Vol 11 (11) ◽  
pp. 1354 ◽  
Author(s):  
Hassan Almusawa ◽  
Ryad Ghanam ◽  
Gerard Thompson
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Vector Fields ◽  
Solvable Lie Groups ◽  
Related System ◽  
Geodesic Equations ◽  
Solvable Lie Algebras ◽  
Symmetry Algebras

In this investigation, we present symmetry algebras of the canonical geodesic equations of the indecomposable solvable Lie groups of dimension five, confined to algebras A 5 , 7 a b c to A 18 a . For each algebra, the related system of geodesics is provided. Moreover, a basis for the associated Lie algebra of the symmetry vector fields, as well as the corresponding nonzero brackets, are constructed and categorized.

A note on the algebra of Poisson brackets

1984 ◽  
Vol 96 (1) ◽  
pp. 45-60 ◽  
Author(s):  
C. J. Atkin
Keyword(s):  
Poisson Bracket ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Symplectic Manifold ◽  
Vector Fields ◽  
Poisson Brackets ◽  
Infinite Dimensional

In a long sequence of notes in the Comptes Rendus and elsewhere, and in the papers [1], [2], [3], [6], [7], Lichnerowicz and his collaborators have studied the ‘classical infinite-dimensional Lie algebras’, their derivations, automorphisms, co-homology, and other properties. The most familiar of these algebras is the Lie algebra of C∞ vector fields on a C∞ manifold. Another is the Lie algebra of ‘Poisson brackets’, that is, of C∞ functions on a C∞ symplectic manifold, with the Poisson bracket as composition; some questions concerning this algebra are of considerable interest in the theory of quantization – see, for instance, [2] and [3].

scholarly journals RESIDUAL SYMMETRIES IN THE PRESENCE OF AN EM BACKGROUND

2003 ◽  
Vol 18 (09) ◽  
pp. 629-641 ◽  
Author(s):  
H. L. CARRION ◽  
M. ROJAS ◽  
F. TOPPAN
Keyword(s):  
Lie Algebra ◽  
Symmetry Algebra ◽  
Algebraic Model ◽  
Two Dimensional ◽  
Square Roots ◽  
Residual Symmetry ◽  
Solvable Lie Algebra ◽  
Poincaré Algebra ◽  
Model Independent ◽  
Symmetry Algebras

The symmetry algebra of a QFT in the presence of an external EM background (named "residual symmetry") is investigated within a Lie-algebraic, model-independent scheme. Some results previously encountered in the literature are extended here. In particular we compute the symmetry algebra for a constant EM background in D = 3 and D = 4 dimensions. In D = 3 dimensions the residual symmetry algebra, for generic values of the constant EM background, is isomorphic to [Formula: see text], with [Formula: see text] the centrally extended two-dimensional Poincaré algebra. In D = 4 dimension the generic residual symmetry algebra is given by a seven-dimensional solvable Lie algebra which is explicitly computed. Residual symmetry algebras are also computed for specific non-constant EM backgrounds and in the supersymmetric case for a constant EM background. The supersymmetry generators are given by the "square roots" of the deformed translations.

Integrable generators of Lie algebras of vector fields on ℂ n

2019 ◽  
Vol 31 (4) ◽  
pp. 943-949
Author(s):  
Rafael B. Andrist
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Vector Fields ◽  
Analogous Result ◽  
Algebraic Vector ◽  
Polynomial Flows

Abstract There exist three vector fields with complete polynomial flows on {\mathbb{C}^{n}} , {n\geq 2} , which generate the Lie algebra generated by all algebraic vector fields on {\mathbb{C}^{n}} with complete polynomial flows. In particular, the flows of these vector fields generate a group that acts infinitely transitively. The analogous result holds in the holomorphic setting.

Cohomology of the vector fields lie algebras on ℝℙ1 acting on bilinear differential operators

2019 ◽  
Vol 56 (3) ◽  
pp. 280-296
Author(s):  
Abdaoui Meher
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Vector Fields ◽  
Differential Operators ◽  
Smooth Vector ◽  
Linear Differential Operators ◽  
Relative Cohomology ◽  
Linear Differential

Abstract Let Vect (ℝℙ1) be the Lie algebra of smooth vector fields on ℝℙ1. In this paper, we classify -invariant linear differential operators from Vect (ℝℙ1) to vanishing on , where is the space of bilinear differential operators acting on weighted densities. This result allows us to compute the first differential -relative cohomology of Vect (ℝℙ1) with coefficients in .

scholarly journals Some lie algebras of vector fields and their derivations Case of partially classical type

1981 ◽  
Vol 82 ◽  
pp. 175-207 ◽  
Author(s):  
Yukihiro Kanie
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Vector Fields ◽  
Tangent Vector ◽  
Classical Type ◽  
Foliated Manifold

Let be a smooth foliated manifold, and the Lie algebra of all leaf-tangent vector fields on M.

Cohomology of the vector fields Lie algebras on ℝ acting on trilinear differential operators, vanishing on 𝔞𝔣𝔣(1)

2017 ◽  
Vol 14 (10) ◽  
pp. 1750150
Author(s):  
Imed Basdouri ◽  
Elamine Nasri ◽  
Hassen Mechi
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Cohomology Group ◽  
Vector Fields ◽  
Differential Operators ◽  
Main Topic ◽  
Smooth Vector ◽  
Relative Cohomology ◽  
Tensor Densities

The main topic of this paper is to compute the first relative cohomology group of the Lie algebra of smooth vector fields [Formula: see text], with coefficients in the space of trilinear differential operators that act on tensor densities, [Formula: see text], vanishing on the Lie algebra [Formula: see text].

scholarly journals $\vee$ -Systems, Holonomy Lie Algebras, and Logarithmic Vector Fields

2017 ◽  
Vol 2018 (7) ◽  
pp. 2070-2098 ◽  
Author(s):  
Misha V Feigin ◽  
Alexander P Veselov
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Special Class ◽  
Vector Fields ◽  
Hyperplane Arrangement ◽  
Gradient Vector ◽  
Coxeter Systems ◽  
Flat Coordinates

Abstract It is shown that the description of certain class of representations of the holonomy Lie algebra $\mathfrak g_{\Delta}$ associated with hyperplane arrangement $\Delta$ is essentially equivalent to the classification of $\vee$-systems associated with $\Delta.$ The flat sections of the corresponding $\vee$-connection can be interpreted as vector fields, which are both logarithmic and gradient. We conjecture that the hyperplane arrangement of any $\vee$-system is free in Saito's sense and show this for all known $\vee$-systems and for a special class of $\vee$-systems called harmonic, which includes all Coxeter systems. In the irreducible Coxeter case the potentials of the corresponding gradient vector fields turn out to be Saito flat coordinates, or their one-parameter deformations. We give formulas for these deformations as well as for the potentials of the classical families of harmonic $\vee$-systems.

COHOMOLOGY OF THE VECTOR FIELDS LIE ALGEBRAS ON ℝℙ1 ACTING ON BILINEAR DIFFERENTIAL OPERATORS

2005 ◽  
Vol 02 (01) ◽  
pp. 23-40 ◽  
Author(s):  
SOFIANE BOUARROUDJ
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Cohomology Group ◽  
Vector Fields ◽  
Differential Operators ◽  
Projective Line ◽  
Smooth Vector ◽  
Relative Cohomology ◽  
First Cohomology Group ◽  
Tensor Densities

The main topic of this paper is two-fold. First, we compute the first relative cohomology group of the Lie algebra of smooth vector fields on the projective line, Vect(ℝℙ1), with coefficients in the space of bilinear differential operators that act on tensor densities, [Formula: see text], vanishing on the Lie algebra sl(2, ℝ). Second, we compute the first cohomology group of the Lie algebra sl(2, ℝ) with coefficients in [Formula: see text].

Vector Fields and Automorphism Groups of Danielewski Surfaces

2020 ◽  
Author(s):  
Matthias Leuenberger ◽  
Andriy Regeta
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Vector Fields ◽  
Automorphism Groups ◽  
Locally Nilpotent

Abstract In this paper, we study the Lie algebra of vector fields ${\operatorname{Vec}}(\textrm{D}_p)$ of a smooth Danielewski surface $\textrm{D}_p$. We prove that the Lie subalgebra $\langle{\operatorname{LNV}}(\textrm{D}_p) \rangle$ of ${\operatorname{Vec}}(\textrm{D}_p)$ generated by locally nilpotent vector fields is simple. Moreover, if the two Lie algebras $\langle{\operatorname{LNV}}(\textrm{D}_p) \rangle$ and $\langle{\operatorname{LNV}}(\textrm{D}_q) \rangle$ of two Danielewski surfaces $\textrm{D}_p$ and $\textrm{D}_q$ are isomorphic, then the surfaces $\textrm{D}_p$ and $\textrm{D}_q$ are isomorphic. As an application we prove that the ind-groups ${\operatorname{Aut}}(\textrm{D}_p)$ and ${\operatorname{Aut}}(\textrm{D}_q)$ are isomorphic if and only if $\textrm{D}_p \simeq \textrm{D}_q$ as a variety. We also show that any automorphism of the ind-group ${\operatorname{Aut}}^\circ (\textrm{D}_p)$ is inner.

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