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.
Read more about this topic: Bracket (mathematics)
Famous quotes containing the word lie:
“Nature herself has not provided the most graceful end for her creatures. What becomes of all these birds that people the air and forest for our solacement? The sparrow seems always chipper, never infirm. We do not see their bodies lie about. Yet there is a tragedy at the end of each one of their lives. They must perish miserably; not one of them is translated. True, “not a sparrow falleth to the ground without our Heavenly Father’s knowledge,” but they do fall, nevertheless.”
—Henry David Thoreau (1817–1862)