scholarly journals Tensor fields associated with integrable systems of chiral

2019 ◽  
Vol 21 (4) ◽  
pp. 405-412
Author(s):  
Alexander V. Balandin
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Necessary Conditions ◽  
Differential Forms ◽  
Type Systems ◽  
Lax Representation ◽  
Killing Tensor ◽  
Tensor Fields ◽  
Linear Differential ◽  
Special Case

This article describes necessary conditions for chiral-type systems to admit Lax representation with values in simple compact Lie algebras. These conditions state that there exists a covariant constant tensor field with an additional property. It is proposed to construct in an invariant way some covariant tensor fields using the Lax representation of the system under consideration. These fields are constructed by taking linear differential forms with values in the Lie algebra that are constructed using the Lax representation of the system and substituting them into an arbitrary Ad-invariant form on the Lie algebra. The paper proves that such tensor fields are Killing tensor fields or covariant constant fields. The discovered necessary conditions for the existence of the Lax representation are obtained using a special case of such tensor fields associated with the Killing metric of the Lie algebra.

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 .

XIX.—Metrisable Lie Groups and Algebras

1957 ◽  
Vol 64 (3) ◽  
pp. 290-304 ◽  
Author(s):  
S.-T. Tsou ◽  
A. G. Walker
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Lie Group ◽  
Necessary Conditions ◽  
Existence Theorems ◽  
Riemannian Metric ◽  
Abelian Algebra ◽  
Lie Groups And Algebras ◽  
Complex Extension

A Lie group is said to be metrisable if it admits a Riemannian metric which is invariant under all translations of the group. It is shown that the study of such groups reduces to the study of what are called metrisable Lie algebras, and some necessary conditions for a Lie algebra to be metrisable are given. Various decomposition and existence theorems are also given, and it is shown that every metrisable algebra is the product of an abelian algebra and a number of non-decomposable reduced algebras. The number of independent metrics admitted by a metrisable algebra is examined, and it is shown that the metric is unique when and only when the complex extension of the algebra is simple.

scholarly journals Generalized Reynolds operators on 3-Lie algebras and NS-3-Lie algebras

2021 ◽  
Author(s):  
Shuai Hou ◽  
Yunhe Sheng
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Cohomology Group ◽  
Algebra Structure ◽  
Algebraic Structures ◽  
Reynolds Operator ◽  
Infinitesimal Deformations ◽  
Special Case

In this paper, first, we introduce the notion of a generalized Reynolds operator on a [Formula: see text]-Lie algebra [Formula: see text] with a representation on [Formula: see text]. We show that a generalized Reynolds operator induces a 3-Lie algebra structure on [Formula: see text], which represents on [Formula: see text]. By this fact, we define the cohomology of a generalized Reynolds operator and study infinitesimal deformations of a generalized Reynolds operator using the second cohomology group. Then we introduce the notion of an NS-[Formula: see text]-Lie algebra, which produces a 3-Lie algebra with a representation on itself. We show that a generalized Reynolds operator induces an NS-[Formula: see text]-Lie algebra naturally. Thus NS-[Formula: see text]-Lie algebras can be viewed as the underlying algebraic structures of generalized Reynolds operators on [Formula: see text]-Lie algebras. Finally, we show that a Nijenhuis operator on a 3-Lie algebra gives rise to a representation of the deformed 3-Lie algebra and a 2-cocycle. Consequently, the identity map will be a generalized Reynolds operator on the deformed 3-Lie algebra. We also introduce the notion of a Reynolds operator on a [Formula: see text]-Lie algebra, which can serve as a special case of generalized Reynolds operators on 3-Lie algebras.

scholarly journals A Note on Exponents vs Root Heights for Complex Simple Lie Algebras

10.37236/1160 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Sankaran Viswanath
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Fourier Coefficients ◽  
Combinatorial Proof ◽  
Simple Lie Algebras ◽  
Simple Lie Algebra ◽  
Positive Roots ◽  
Special Case

We give an elementary combinatorial proof of a special case of a result due to Bazlov and Ion concerning the Fourier coefficients of the Cherednik kernel. This can be used to give yet another proof of the classical fact that for a complex simple Lie algebra $\mathfrak{ g}$, the partition formed by the exponents of $\mathfrak{ g}$ is dual to that formed by the numbers of positive roots at each height.

scholarly journals COVARIANT STAR PRODUCT ON SYMPLECTIC AND POISSON SPACE–TIME MANIFOLDS

2010 ◽  
Vol 25 (18n19) ◽  
pp. 3765-3796 ◽  
Author(s):  
M. CHAICHIAN ◽  
M. OKSANEN ◽  
A. TUREANU ◽  
G. ZET
Keyword(s):  
Deformation Quantization ◽  
Star Product ◽  
Differential Forms ◽  
Linear Connection ◽  
Poisson Manifold ◽  
Star Products ◽  
Tensor Fields ◽  
Smooth Functions ◽  
Poisson Space ◽  
Special Case

A covariant Poisson bracket and an associated covariant star product in the sense of deformation quantization are defined on the algebra of tensor-valued differential forms on a symplectic manifold, as a generalization of similar structures that were recently defined on the algebra of (scalar-valued) differential forms. A covariant star product of arbitrary smooth tensor fields is obtained as a special case. Finally, we study covariant star products on a more general Poisson manifold with a linear connection, first for smooth functions and then for smooth tensor fields of any type. Some observations on possible applications of the covariant star products to gravity and gauge theory are made.

Integrable two-dimensional dynamical systems and the characteristic Lie algebras

2007 ◽  
Vol 5 ◽  
pp. 195-200
Author(s):  
A.V. Zhiber ◽  
O.S. Kostrigina
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Second Order ◽  
System Of Equations ◽  
Two Dimensional ◽  
Finite Dimensional ◽  
Dimensional Dynamical System

In the paper it is shown that the two-dimensional dynamical system of equations is Darboux integrable if and only if its characteristic Lie algebra is finite-dimensional. The class of systems having a full set of fist and second order integrals is described.

Reduction of partially invariant submodels of rank 3 defect 1 to invariant submodels

2018 ◽  
Vol 13 (3) ◽  
pp. 59-63 ◽  
Author(s):  
D.T. Siraeva
Keyword(s):  
Equation Of State ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Gas Dynamics ◽  
Hydrodynamic Type ◽  
System Of Equations ◽  
Two Dimensional ◽  
Equations Of Gas Dynamics ◽  
Entropy Functions

Equations of hydrodynamic type with the equation of state in the form of pressure separated into a sum of density and entropy functions are considered. Such a system of equations admits a twelve-dimensional Lie algebra. In the case of the equation of state of the general form, the equations of gas dynamics admit an eleven-dimensional Lie algebra. For both Lie algebras the optimal systems of non-similar subalgebras are constructed. In this paper two partially invariant submodels of rank 3 defect 1 are constructed for two-dimensional subalgebras of the twelve-dimensional Lie algebra. The reduction of the constructed submodels to invariant submodels of eleven-dimensional and twelve-dimensional Lie algebras is proved.

scholarly journals Structure of n-Lie Algebras with Involutive Derivations

2018 ◽  
Vol 2018 ◽  
pp. 1-9
Author(s):  
Ruipu Bai ◽  
Shuai Hou ◽  
Yansha Gao
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Two Dimensional

We study the structure of n-Lie algebras with involutive derivations for n≥2. We obtain that a 3-Lie algebra A is a two-dimensional extension of Lie algebras if and only if there is an involutive derivation D on A=A1  ∔  A-1 such that dim A1=2 or dim A-1=2, where A1 and A-1 are subspaces of A with eigenvalues 1 and -1, respectively. We show that there does not exist involutive derivations on nonabelian n-Lie algebras with n=2s for s≥1. We also prove that if A is a (2s+2)-dimensional (2s+1)-Lie algebra with dim A1=r, then there are involutive derivations on A if and only if r is even, or r satisfies 1≤r≤s+2. We discuss also the existence of involutive derivations on (2s+3)-dimensional (2s+1)-Lie algebras.

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