Bracket (mathematics) - Lie Bracket and Commutator

Lie Bracket and Commutator

In group theory and ring theory, square brackets are used to denote the commutator. In group theory, the commutator is commonly defined as g−1h−1gh. In ring theory, the commutator is defined as ab − ba. Furthermore, in ring theory, braces are used to denote the anticommutator where {a,b} is defined as ab + ba.

The Lie bracket of a Lie algebra is a binary operation denoted by . By using the commutator as a Lie bracket, every associative algebra can be turned into a Lie algebra. There are many different forms of Lie bracket, in particular the Lie derivative and the Jacobi-Lie bracket.