Fundamental Vector Fields

2020 ◽  
pp. 87-96
Author(s):  
Loring W. Tu
Keyword(s):  
Vector Field ◽  
Fixed Points ◽  
Lie Algebra ◽  
Group Action ◽  
Lie Group ◽  
Vector Fields ◽  
Principal Bundle ◽  
Left Action ◽  
Fundamental Vector ◽  
Vertical Tangent

This chapter addresses fundamental vector fields. The concept of a connection on a principal bundle is essential in the construction of the Cartan model. To define a connection on a principal bundle, one first needs to define the fundamental vector fields. When a Lie group acts smoothly on a manifold, every element of the Lie algebra of the Lie group generates a vector field on the manifold called a fundamental vector field. On a principal bundle, the fundamental vectors are precisely the vertical tangent vectors. In general, there is a relation between zeros of fundamental vector fields and fixed points of the group action. Unless specified otherwise (such as on a principal bundle), a group action is assumed to be a left action.

VECTOR FIELDS, VARIATIONAL EQUATIONS AND COMMUTATORS

2012 ◽  
Vol 22 (08) ◽  
pp. 1250190
Author(s):  
WILLI-HANS STEEB ◽  
YORICK HARDY ◽  
IGOR TANSKI
Keyword(s):  
Vector Field ◽  
Differential Equations ◽  
Fixed Points ◽  
Lie Algebra ◽  
Autonomous System ◽  
Vector Fields ◽  
Autonomous Systems ◽  
First Integrals ◽  
Variational Equations ◽  
First Order

We study autonomous systems of first order ordinary differential equations, their corresponding vector fields and the autonomous system corresponding to the vector field of the commutator of two such autonomous systems. These vector fields form a Lie algebra. From the variational equations of these autonomous systems, we form new vector fields consisting of the sum of the two vector fields. We show that these new vector fields also form a Lie algebra. Results about fixed points, first integrals and the divergence of the vector fields are also presented.

Harish-Chandra Modules over ℤ

2019 ◽  
pp. 126-167
Author(s):  
Günter Harder
Keyword(s):  
Fixed Points ◽  
Lie Algebra ◽  
Lie Group ◽  
Maximal Torus ◽  
Reductive Group ◽  
Group Scheme ◽  
Finitely Generated ◽  
Cartan Involution ◽  
Split Maximal Torus ◽  
Definition Of

This chapter shows that certain classes of Harish-Chandra modules have in a natural way a structure over ℤ. The Lie group is replaced by a split reductive group scheme G/ℤ, its Lie algebra is denoted by 𝖌ℤ. On the group scheme G/ℤ there is a Cartan involution 𝚯 that acts by t ↦ t −1 on the split maximal torus. The fixed points of G/ℤ under 𝚯 is a flat group scheme 𝒦/ℤ. A Harish-Chandra module over ℤ is a ℤ-module 𝒱 that comes with an action of the Lie algebra 𝖌ℤ, an action of the group scheme 𝒦, and some compatibility conditions is required between these two actions. Finally, 𝒦-finiteness is also required, which is that 𝒱 is a union of finitely generated ℤ modules 𝒱I that are 𝒦-invariant. The definitions imitate the definition of a Harish-Chandra modules over ℝ or over ℂ.

scholarly journals On diagrammatic technique for nonlinear dynamical systems

2014 ◽  
Vol 29 (35) ◽  
pp. 1430039
Author(s):  
Mykola Semenyakin
Keyword(s):  
Vector Field ◽  
Dynamical Systems ◽  
Fixed Points ◽  
Vector Fields ◽  
Nonlinear Dynamical Systems ◽  
Radius Of Convergence ◽  
Evolution Operators ◽  
Finite Degree ◽  
Nonlinear Dynamical ◽  
Diagrammatic Technique

In this paper, we investigate phase flows over ℂn and ℝn generated by vector fields V = ∑ Pi∂i where Pi are finite degree polynomials. With the convenient diagrammatic technique, we get expressions for evolution operators ev {V|t} : x(0) ↦ x(t) through the series in powers of x(0) and t, represented as sum over all trees of a particular type. Estimates are made for the radius of convergence in some particular cases. The phase flows behavior in the neighborhood of vector field fixed points are examined. Resonance cases are considered separately.

scholarly journals Ruang Fase Tereduksi Grup Lie Aff (1)

2021 ◽  
Vol 3 (2) ◽  
pp. 180-186
Author(s):  
Edi Kurniadi
Keyword(s):  
Phase Space ◽  
Lie Algebra ◽  
Group Action ◽  
Lie Group ◽  
Dual Space ◽  
Coadjoint Orbits ◽  
Coadjoint Representation

ABSTRAKDalam artikel ini dipelajari ruang fase tereduksi dari suatu grup Lie khususnya untuk grup Lie affine  berdimensi 2. Tujuannya adalah untuk mengidentifikasi ruang fase tereduksi dari  melalui orbit coadjoint buka di titik tertentu pada ruang dual  dari aljabar Lie . Aksi dari grup Lie    pada ruang dual  menggunakan representasi coadjoint. Hasil yang diperoleh adalah ruang Fase tereduksi  tiada lain adalah orbit coadjoint-nya yang buka di ruang dual . Selanjutnya, ditunjukkan pula bahwa grup Lie affine     tepat mempunyai dua buah orbit coadjoint buka.  Hasil yang diperoleh dalam penelitian ini dapat diperluas untuk kasus grup Lie affine  berdimensi  dan untuk kasus grup Lie lainnya.ABSTRACTIn this paper, we study a reduced phase space for a Lie group, particularly for the 2-dimensional affine Lie group which is denoted by Aff (1). The work aims to identify the reduced phase space for Aff (1) by open coadjoint orbits at certain points in the dual space aff(1)* of the Lie algebra aff(1). The group action of Aff(1) on the dual space aff(1)* is considered using coadjoint representation. We obtained that the reduced phase space for the affine Lie group Aff(1) is nothing but its open coadjoint orbits. Furthermore, we show that the affine Lie group Aff (1) exactly has two open coadjoint orbits in aff(1)*. Our result can be generalized for the n(n+1) dimensional affine Lie group Aff(n) and for another Lie group.

scholarly journals A Lie connection between Hamiltonian and Lagrangian optics

1997 ◽  
Vol Vol. 1 ◽  
Author(s):  
Alex J. Dragt
Keyword(s):  
Vector Field ◽  
Lie Algebra ◽  
Vector Fields ◽  
Geometrical Optics ◽  
Hamiltonian Vector ◽  
Hamiltonian Vector Field ◽  
Third Order ◽  
Hamiltonian Vector Fields ◽  
Uniform Medium ◽  
International Audience

International audience It is shown that there is a non-Hamiltonian vector field that provides a Lie algebraic connection between Hamiltonian and Lagrangian optics. With the aid of this connection, geometrical optics can be formulated in such a way that all aberrations are attributed to ray transformations occurring only at lens surfaces. That is, in this formulation there are no aberrations arising from simple transit in a uniform medium. The price to be paid for this formulation is that the Lie algebra of Hamiltonian vector fields must be enlarged to include certain non-Hamiltonian vector fields. It is shown that three such vector fields are required at the level of third-order aberrations, and sufficient machinery is developed to generalize these results to higher order.

scholarly journals The Lie Group in Infinite Dimension

2011 ◽  
Vol 2011 ◽  
pp. 1-35 ◽  
Author(s):  
V. Tryhuk ◽  
V. Chrastinová ◽  
O. Dlouhý
Keyword(s):  
Lie Algebra ◽  
Lie Group ◽  
Vector Fields ◽  
Dimensional Space ◽  
Finite Dimensional Space ◽  
Infinite Dimensional ◽  
Symmetries Of Differential Equations ◽  
Smooth Action ◽  
Finite Dimensional ◽  
Infinitesimal Transformations

A Lie group acting on finite-dimensional space is generated by its infinitesimal transformations and conversely, any Lie algebra of vector fields in finite dimension generates a Lie group (the first fundamental theorem). This classical result is adjusted for the infinite-dimensional case. We prove that the (local,C∞smooth) action of a Lie group on infinite-dimensional space (a manifold modelled onℝ∞) may be regarded as a limit of finite-dimensional approximations and the corresponding Lie algebra of vector fields may be characterized by certain finiteness requirements. The result is applied to the theory of generalized (or higher-order) infinitesimal symmetries of differential equations.

scholarly journals The generalized centrally extended Lie algebraic structures and related integrable heavenly type equations

2020 ◽  
Vol 12 (1) ◽  
pp. 242-264
Author(s):  
O.Ye. Hentosh ◽  
A.A. Balinsky ◽  
A.K. Prykarpatski
Keyword(s):  
Vector Field ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Wide Class ◽  
Vector Fields ◽  
Arbitrary Dimension ◽  
Multidimensional Case ◽  
Coadjoint Orbits ◽  
Algebraic Structures ◽  
Adjoint Space

There are studied Lie-algebraic structures of a wide class of heavenly type non-linear integrable equations, related with coadjoint flows on the adjoint space to a loop vector field Lie algebra on the torus. These flows are generated by the loop Lie algebras of vector fields on a torus and their coadjoint orbits and give rise to the compatible Lax-Sato type vector field relationships. The related infinite hierarchy of conservations laws is analysed and its analytical structure, connected with the Casimir invariants, is discussed. We present the typical examples of such equations and demonstrate in details their integrability within the scheme developed. As examples, we found and described new multidimensional generalizations of the Mikhalev-Pavlov and Alonso-Shabat type integrable dispersionless equation, whose seed elements possess a special factorized structure, allowing to extend them to the multidimensional case of arbitrary dimension.

scholarly journals LIFTING HAMILTONIAN LOOPS TO ISOTOPIES IN FIBRATIONS

2013 ◽  
Vol 10 (10) ◽  
pp. 1350057
Author(s):  
ANDRÉS VIÑA
Keyword(s):  
Vector Field ◽  
Lie Group ◽  
Closed Subgroup ◽  
Principal Bundle ◽  
Sufficient Condition ◽  
Flag Varieties ◽  
Toric Manifold ◽  
Mass Center ◽  
Lattice Vector ◽  
Hamiltonian Groups

Let G be a Lie group, H a closed subgroup and M the homogeneous space G/H. Each representation Ψ of H determines a G-equivariant principal bundle [Formula: see text] on M endowed with a G-invariant connection. We consider subgroups [Formula: see text] of the diffeomorphism group Diff (M), such that, each vector field [Formula: see text] admits a lift to a preserving connection vector field on [Formula: see text]. We prove that [Formula: see text]. This relation is applicable to subgroups [Formula: see text] of the Hamiltonian groups of the flag varieties of a semisimple group G. Let MΔ be the toric manifold determined by the Delzant polytope Δ. We put φb for the loop in the Hamiltonian group of MΔ defined by the lattice vector b. We give a sufficient condition, in terms of the mass center of Δ, for the loops φb and [Formula: see text] to be homotopically inequivalent.

scholarly journals Hom-Lie Algebras and Hom-Lie Groups, Integration and Differentiation

2020 ◽  
Author(s):  
Jun Jiang ◽  
◽  
Satyendra Kumar Mishra ◽  
Yunhe Sheng ◽  
◽  
...  
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Group Action ◽  
Lie Group ◽  
Smooth Manifold ◽  
Adjoint Representation ◽  
Lie Group Action

In this paper, we introduce the notion of a (regular) Hom-Lie group. We associate a Hom-Lie algebra to a Hom-Lie group and show that every regular Hom-Lie algebra is integrable. Then, we define a Hom-exponential (Hexp) map from the Hom-Lie algebra of a Hom-Lie group to the Hom-Lie group and discuss the universality of this Hexp map. We also describe a Hom-Lie group action on a smooth manifold. Subsequently, we give the notion of an adjoint representation of a Hom-Lie group on its Hom-Lie algebra. At last, we integrate the Hom-Lie algebra (gl(V),[.,.],Ad), and the derivation Hom-Lie algebra of a Hom-Lie algebra.

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