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.
リー括弧は、[X, Y]と表され、ここでXとYはリー代数の要素です。これは、要素がどれだけ非可換であるかを測定します。具体的には、反対称性を持ち、[X, Y] = -[Y, X]となり、ヤコビ恒等式は、任意の要素X、Y、Zに対して、[X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y]] = 0を満たすことを意味します。
リーマン代数は、表現論を含むさまざまな分野で重要です、 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 量子力学 そしてゲージ理論の定式化にも関係しています。
一般的なリーマン代数の例には、特別なものがあります 線形代数 (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.