Symmetry of Lie Algebras

Dear Colleagues,

The most powerful methods for solving differential equations are symmetry-based methods. These methods originated from the Lie method, which was created by the prominent Norwegian mathematician, Sophus Lie. His theory and method have continuously been in focus of research of many well-known mathematicians, computer scientists and physicists. Although the technique of the Lie method is well-known, the method still attracts the attention of many researchers, and new results are published on a regular basis. This Special Issue is devoted to recent development of Lie theory and its applications for solving physically and biologically motivated equations and models. In particular, the issue welcomes articles devoted to analysis and classification of Lie algebras, which are invariance algebras of real world models; Lie and conditional symmetry classification problems of nonlinear PDEs; the application of symmetry based methods for finding new exact solutions of nonlinear PDEs arising in applications. Articles and reviews devoted to the theoretical foundations of symmetry based methods and their applications for solving other nonlinear and nonlinear models are also welcome.

In analysis and classification of Lie algebras, derivations and homomorphisms in Lie algebras play important roles. The construction of symmetric and anti-symmetric bi-derivations and bi-homomorphisms in Lie algebra containing Lie C*-algebras are devoted to understanding the structure of Lie algebras. In the analysis, the Lie symmetry analysis method has been proposed for finding similarity reduction and exact solutions of nonlinear evolution equation. Also by using symmetry reduction method, nonlinear partial differential equation to nonlinear ordinary differential equation, which has advantage to provide semi analytical solution, can be reduced. By using Lie symmetry analysis method, infinitesimal generators, the entire geometric vector field, commutator table of Lie algebra and symmetry group can be obtained. The new concept of nonlinear self-adjointness of differential equations is used for construction of nonlocal conservation laws. The conservation laws for the some equations can be obtained by using the new conservation theorem method, the formal Lagrangian approach and the Lie symmetry operators.

For this Special Issue, we would like to invite original research and review articles that (i) focus on and highlight the Lie symmetry method in the investigation of differential equations, that (ii) center to recent development of Lie theory and its applications for solving physically and biologically motivated equations and models or that (iii) concentrate on symmetric and anti-symmetric bi-derivations and bi-homomorphisms in Lie algebras.

Prof. Dr. Themistocles M. Rassias
Guest Editors

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• Lie symmetry
• invariance algebra of nonlinear PDE
• symmetry of (initial) boundary-value problem
• Lie algebra
• representation of Lie algebra
• conditional symmetry
• nonlinear boundary-value problem
• Lie symmetry analysis method
• Lie C*-algebra
• symmetric bi-homomorphism
• symmetric bi-derivation
• symmetric reduction
• geometric approach