scholarly journals A Basis of Casimirs in 3D Magnetohydrodynamics

2020 ◽  
Author(s):  
Boris Khesin ◽  
Daniel Peralta-Salas ◽  
Cheng Yang
Keyword(s):  
Lie Algebra ◽  
Semidirect Product ◽  
Dual Space ◽  
Magnetic Helicity ◽  
Coadjoint Action ◽  
Integral Invariants ◽  
Volume Preserving

Abstract We prove that any regular Casimir in 3D magnetohydrodynamics (MHD) is a function of the magnetic helicity and cross-helicity. In other words, these two helicities are the only independent regular integral invariants of the coadjoint action of the MHD group $\textrm{SDiff}(M)\ltimes \mathfrak X^*(M)$, which is the semidirect product of the group of volume-preserving diffeomorphisms and the dual space of its Lie algebra.

scholarly journals A Drift-Free Left Invariant Control System on the Lie Group SO(3)×R3×R3

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Camelia Pop
Keyword(s):  
Control System ◽  
Numerical Integration ◽  
Lie Algebra ◽  
Lie Group ◽  
Poisson Structure ◽  
Dual Space ◽  
Free System ◽  
Geometrical Properties ◽  
Free Left ◽  
Invariant Control

A controllable drift-free system on the Lie group G=SO(3)×R3×R3 is considered. The dynamics and geometrical properties of the corresponding reduced Hamilton’s equations on g∗,·,·- are studied, where ·,·- is the minus Lie-Poisson structure on the dual space g∗ of the Lie algebra g=so(3)×R3×R3 of G. The numerical integration of this system is also discussed.

EXTENDED BOTT–VIRASORO ALGEBRA, SEMIDIRECT PRODUCT, ⋆-LIE ALGEBRA OF DIFFEOMORPHISM AND NONCOMMUTATIVE INTEGRABLE SYSTEMS

2009 ◽  
Vol 06 (04) ◽  
pp. 555-572
Author(s):  
PARTHA GUHA
Keyword(s):  
Lie Algebra ◽  
Semidirect Product ◽  
Virasoro Algebra ◽  
Lie Derivative ◽  
Coadjoint Representation ◽  
Kdv Equations ◽  
Two Component ◽  
Universal Extension ◽  
Product Algebra ◽  
Tensor Densities

We study noncommutative theory of a coadjoint representation of a universal extension of Vect (S1) ⋉ C∞(S1) algebra using the action of ⋆-deformed matrix Hill's operators Δ⋆ on the space of ⋆-deformed tensor densities. The centrally extended semidirect product algebra [Formula: see text] is a Lie algebra of extended semidirect product of the Bott–Virasoro group [Formula: see text]. The study of deformed diffeomorphisms, deformed semidirect product algebra and deformed Lie derivative action of Δ⋆ on ⋆ deformed tensor-densities on S1 allow us to construct noncommutative two component Korteweg–de Vries (KdV) equations, in particular, we derive the noncommutative Ito equation. This leads to a geometric formulation of ⋆-deformed quantization of the centrally extended semidirect product algebra [Formula: see text] and two component noncommutative KdV equations.

On basic Cycles of An, Bn, Cn and Dn

1985 ◽  
Vol 37 (1) ◽  
pp. 122-140 ◽  
Author(s):  
D. J. Britten ◽  
F. W. Lemire
Keyword(s):  
Lie Algebra ◽  
Dual Space ◽  
Cartan Subalgebra ◽  
Closed Field ◽  
Generating Set ◽  
Enveloping Algebra ◽  
Simple Lie Algebra ◽  
Finite Dimensional ◽  
Irreducible Modules ◽  
Positive Roots

In this paper, we investigate a conjecture of Dixmier [2] on the structure of basic cycles. Our interest in basic cycles arises primarily from the fact that the irreducible modules of a simple Lie algebra L having a weight space decomposition are completely determined by the irreducible modules of the cycle subalgebra of L. The basic cycles form a generating set for the cycle subalgebra.First some notation: F denotes an algebraically closed field of characteristic 0, L a finite dimensional simple Lie algebra of rank n over F, H a fixed Cartan subalgebra, U(L) the universal enveloping algebra of L, C(L) the centralizer of H in U(L), Φ the set of nonzero roots in H*, the dual space of H, Δ = {α1, …, αn} a base of Φ, and Φ+ = {β1, …, βm} the positive roots corresponding to Δ.

Invariants in dissipationless hydrodynamic media

1993 ◽  
Vol 248 ◽  
pp. 67-106 ◽  
Author(s):  
A. V. Tur ◽  
V. V. Yanovsky
Keyword(s):  
Lie Algebra ◽  
Vector Fields ◽  
Infinite Family ◽  
Type Systems ◽  
Topological Invariant ◽  
Systems Of Equations ◽  
Integral Invariants ◽  
New Type ◽  
General Relations ◽  
Dynamic Invariants

We propose a general geometric method of derivation of invariant relations for hydrodynamic dissipationless media. New dynamic invariants are obtained. General relations between the following three types of invariants are established, valid in all models: Lagrangian invariants, frozen-in vector fields and frozen-in co-vector fields. It is shown that frozen-in integrals form a Lie algebra with respect to the commutator of the frozen fields. The relation between frozen-in integrals derived here can be considered as the Backlund transformation for hydrodynamic-type systems of equations. We derive an infinite family of integral invariants which have either dynamic or topological nature. In particular, we obtain a new type of topological invariant which arises in all hydrodynamic dissipationless models when the well-known Moffatt invariant vanishes.

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 Tθ∗-Extensions of n-Lie Algebras

ISRN Algebra ◽  
2011 ◽  
Vol 2011 ◽  
pp. 1-11
Author(s):  
Ruipu Bai ◽  
Ying Li
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Dual Space

The paper is mainly concerned with Tθ∗-extensions of n-Lie algebras. The Tθ∗-extension Lθ(L∗) of an n-Lie algebra L by a cocycle θ is defined, and a class of cocycles is constructed by means of linear mappings from an n-Lie algebra on to its dual space. Finally all Tθ∗-extensions of (n+1)-dimensional n-Lie algebras are classified, and the explicit multiplications are given.

scholarly journals DEFORMATION QUANTIZATION OF CERTAIN NONLINEAR POISSON STRUCTURES

1998 ◽  
Vol 09 (05) ◽  
pp. 599-621 ◽  
Author(s):  
BYUNG-JAY KAHNG
Keyword(s):  
Poisson Bracket ◽  
Lie Algebra ◽  
Deformation Quantization ◽  
Dual Space ◽  
Poisson Brackets ◽  
Poisson Structures ◽  
Central Extensions ◽  
C Algebras ◽  
Lie Bracket ◽  
Compact Quantum Groups

As a generalization of the linear Poisson bracket on the dual space of a Lie algebra, we introduce certain nonlinear Poisson brackets which are "cocycle perturbations" of the linear Poisson bracket. We show that these special Poisson brackets are equivalent to Poisson brackets of central extension type, which resemble the central extensions of an ordinary Lie bracket via Lie algebra cocycles. We are able to formulate (strict) deformation quantizations of these Poisson brackets by means of twisted group C*-algebras. We also indicate that these deformation quantizations can be used to construct some specific non-compact quantum groups.

scholarly journals Representasi Unitar Tak Tereduksi Grup Lie Dari Aljabar Lie Filiform Real Berdimensi 5

2020 ◽  
Vol 17 (1) ◽  
pp. 100-108
Author(s):  
E Kurniadi
Keyword(s):  
Harmonic Analysis ◽  
Lie Algebra ◽  
Unitary Representation ◽  
Lie Group ◽  
Dual Space ◽  
Irreducible Unitary Representation ◽  
Coadjoint Orbits ◽  
Dimensional Representation ◽  
Induced Representation ◽  
Filiform Lie Algebra

In this paper, we study a harmonic analysis of a Lie group  of a real filiform Lie algebra of dimension 5. Particularly, we study its  irreducible unitary representation (IUR) and contruct this IUR corresponds to its coadjoint orbits through coadjoint actions of its group to its dual space.  Using induced representation of  a 1-dimensional representation of its subgroup we obtain its IUR of its Lie group

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