Zeitschrift für verallgemeinerte Lügentheorie und -anwendungen

Advancements in Nonlinear Lie Theory and its Implications for Algebraic Structures

Abstract

Chiotellis Andreia*

Nonlinear Lie theory is an area of mathematics that explores the extension of Lie algebras and Lie groups into nonlinear contexts, offering profound implications for various algebraic structures. Traditionally, Lie theory has been a cornerstone in understanding continuous symmetries through linear transformations, particularly in physics and geometry. However, the evolution of nonlinear Lie theory expands these ideas to more complex systems where linearity is no longer a given, leading to new insights and applications across diverse mathematical fields. At its core, Lie theory is concerned with the study of Lie groups, which are groups that also have the structure of a differentiable manifold, and Lie algebras, which are the tangent space at the identity of these groups, equipped with a bilinear operation known as the Lie bracket. Lie groups provide a natural framework for analyzing continuous symmetries, while Lie algebras capture the infinitesimal structure of these groups. The classical theory is deeply rooted in linear structures, where the operations and transformations are linear maps. However, many systems in mathematics and physics exhibit nonlinear behavior, prompting the need to extend Lie theory into this nonlinear regime.