A lie algebra is a fundamental concept in mathematics and physics, specifically in the field of algebraic structures. It is a vector space equipped with a binary operation called Lie bracket that satisfies certain properties.
Formally, a lie algebra is a vector space L over a field F, often taken to be the real numbers or complex numbers, along with a bilinear operation [ , ]: L x L -> L, typically denoted [x, y], which is called the Lie bracket. This operation satisfies three fundamental axioms:
1. Bilinearity: The Lie bracket is linear in both arguments, meaning that for any scalars a, b in F and elements x, y, and z in L, we have [ax + by, z] = a[x, z] + b[y, z] and [z, ax + by] = a[z, x] + b[z, y].
2. Anti-commutativity: The Lie bracket is anti-commutative, meaning that for any elements x and y in L, we have [x, y] = -[y, x].
3. Jacobi identity: The Lie bracket satisfies the Jacobi identity, which states that for any elements x, y, and z in L, we have [[x, y], z] + [[y, z] , x] + [[z, x], y] = 0.
Lie algebras are often used to study symmetry, transformations, and infinitesimal generators in mathematics and physics. They provide a powerful framework for understanding the structure of Lie groups and are essential in many areas such as differential geometry, quantum mechanics, and particle physics.
