Lie algebras are algebraic structures that arise in the study of symmetry and transformations in mathematics and physics. They are named after the Norwegian mathematician Sophus Lie, who developed the theory of continuous symmetry. At their core, Lie algebras consist of a vector space equipped with a binary operation called the Lie bracket, which satisfies two main properties: bilinearity and the Jacobi identity.
Le crochet de Lie, noté [X, Y], où X et Y sont des éléments de l'algèbre de Lie, mesure à quel point ces éléments sont non commutatifs. Plus précisément, il est antisymétrique, ce qui signifie que [X, Y] = -[Y, X], et l'identité de Jacobi stipule que [X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y]] = 0 pour tous éléments X, Y et Z de l'algèbre.
Les algèbres de Lie sont essentielles dans divers domaines, notamment la théorie des représentations, geometry, and theoretical physics. They provide a framework for understanding the algebraic structures underlying continuous transformation groups, such as rotations and translations in space. In physics, Lie algebras play a significant role in the study of symmetries of physical systems, particularly in la mécanique quantique et la formulation des théories de jauge.
Des exemples courants d'algèbres de Lie incluent la spéciale algèbre linéaire (SL(n)), which consists of n×n matrices with determinant equal to one, and the algebra of angular momentum in quantum mechanics. Lie algebras also serve as the foundation for more complex structures, such as Lie groups, which are groups that are also differentiable manifolds, allowing for the study of continuous symmetries and transformations.