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.
O brackets de Lie, denotado como [X, Y], onde X e Y são elementos da álgebra de Lie, mede o quão não comutativos são os elementos. Especificamente, é anti-simétrico, ou seja, [X, Y] = -[Y, X], e a identidade de Jacobi afirma que [X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y]] = 0 para quaisquer elementos X, Y e Z na álgebra.
Álgebras de Lie são cruciais em vários campos, incluindo teoria de representação, 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 mecânica quântica e a formulação de teorias de gauge.
Exemplos comuns de álgebras de Lie incluem a álgebra especial álgebra linear (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.