Group classification of differential-difference equations: Low-dimensional Lie algebras

2017 ◽  
Vol 33 (2) ◽  
pp. 345-362 ◽  
Author(s):  
Shou-feng Shen ◽  
Yong-yang Jin
Keyword(s):  
Lie Algebras ◽  
Difference Equations ◽  
Group Classification ◽  
Low Dimensional

Lie group classification of second-order ordinary difference equations

2000 ◽  
Vol 41 (1) ◽  
pp. 480-504 ◽  
Author(s):  
Vladimir Dorodnitsyn ◽  
Roman Kozlov ◽  
Pavel Winternitz
Keyword(s):  
Lie Group ◽  
Difference Equations ◽  
Group Classification ◽  
Second Order ◽  
Lie Group Classification

Classification of Low-Dimensional Hom-Lie Algebras

2020 ◽  
pp. 223-256
Author(s):  
Elvice Ongong’a ◽  
Johan Richter ◽  
Sergei Silvestrov
Keyword(s):  
Lie Algebras ◽  
Low Dimensional

The classification of three-dimensional Lie algeb­ras on complex field

2020 ◽  
pp. 143-155
Author(s):  
E. R. Shamardina
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Three Dimensional ◽  
Structural Equations ◽  
Complex Numbers ◽  
Complex Field ◽  
Low Dimensional

In this paper, we study the classification of three-dimensional Lie al­gebras over a field of complex numbers up to isomorphism. The proposed classification is based on the consideration of objects invariant with re­spect to isomorphism, namely such quantities as the derivative of a subal­gebra and the center of a Lie algebra. The above classification is distin­guished from others by a more detailed and simple presentation. Any two abelian Lie algebras of the same dimension over the same field are isomorphic, so we understand them completely, and from now on we shall only consider non-abelian Lie algebras. Six classes of three-dimensional Lie algebras not isomorphic to each other over a field of complex numbers are presented. In each of the classes, its properties are described, as well as structural equations defining each of the Lie alge­bras. One of the reasons for considering these low dimensional Lie alge­bras that they often occur as subalgebras of large Lie algebras

scholarly journals Minimal linear representations of the low-dimensional nilpotent Lie algebras

2008 ◽  
Vol 102 (1) ◽  
pp. 17 ◽  
Author(s):  
J. C. Benjumea ◽  
J. Núnez ◽  
A. F. Tenorio
Keyword(s):  
Lie Algebras ◽  
Nilpotent Lie Algebra ◽  
Matrix Representations ◽  
Triangular Matrices ◽  
Nilpotent Lie Algebras ◽  
Linear Representations ◽  
Minimal Dimension ◽  
The Matrix ◽  
Low Dimensional

The main goal of this paper is to compute a minimal matrix representation for each non-isomorphic nilpotent Lie algebra of dimension less than $6$. Indeed, for each of these algebras, we search the natural number $n\in\mathsf{N}\setminus\{1\}$ such that the linear algebra $\mathfrak{g}_n$, formed by all the $n \times n$ complex strictly upper-triangular matrices, contains a representation of this algebra. Besides, we show an algorithmic procedure which computes such a minimal representation by using the Lie algebras $\mathfrak{g}_n$. In this way, a classification of such algebras according to the dimension of their minimal matrix representations is obtained. In this way, we improve some results by Burde related to the value of the minimal dimension of the matrix representations for nilpotent Lie algebras.

Group classification of third order nonlinear wave equations chain’s

2020 ◽  
Vol 15 (3-4) ◽  
pp. 232-237
Author(s):  
O.K. Babkov ◽  
G.Z. Mukhametova
Keyword(s):  
Lie Algebras ◽  
Nonlinear Wave ◽  
Wave Equations ◽  
Group Analysis ◽  
Group Classification ◽  
Nonlinear Wave Equations ◽  
Bäcklund Transformations ◽  
Third Order ◽  
A Chain

The paper presents the results of point symmetrys Lie algebras for third-order nonlinear wave equations calculating linked into a chain by Bäcklund transformations. Calculations are carried out by using Lie-Ovsyannikov method of group analysis. The basic algebras of point symmetries of the indicated equations are found, all possible cases of their extension are revealed, and the commutator tables of algebras found are calculated.

scholarly journals Linear Bundle of Lie Algebras Applied to the Classification of Real Lie Algebras

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1455
Author(s):  
Alina Dobrogowska ◽  
Karolina Wojciechowicz
Keyword(s):  
Lie Algebras ◽  
Hamiltonian Structure ◽  
Three Dimensional ◽  
Poisson Structures ◽  
Triangular Matrices ◽  
Dual Spaces ◽  
Upper Triangular Matrices ◽  
Low Dimensional ◽  
Algebra Level

We present a new look at the classification of real low-dimensional Lie algebras based on the notion of a linear bundle of Lie algebras. Belonging to a suitable family of Lie bundles entails the compatibility of the Lie–Poisson structures with the dual spaces of those algebras. This gives compatibility of bi-Hamiltonian structure on the space of upper triangular matrices and with a bundle at the algebra level. We will show that all three-dimensional Lie algebras belong to two of these families and four-dimensional Lie algebras can be divided in three of these families.

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