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Lie Algebra

Basic definition

A Lie algebra is a nonassociative algebra corresponding to a Lie group.

Detailed definition

A nonassociative algebra obeyed by objects such as the Lie bracket and Poisson bracket. Elements f, g, and h of a Lie algebra satisfy
[f, f] = 0
[f + g, h] = [f, h] + [g, h], and
[f, [g, h]] + [g, [h, f]] + [h, [f, g]] = 0
(the Jacobi identity). The relation [f, f] = 0 implies
[f, g] = - [g, f].
For characteristic not equal to two, these two relations are equivalent.
The binary operation of a Lie algebra is the bracket
[f g, h] = f[g, h] + [f, h] g.

Educational grade level

graduate school level

Associated person

Marius Sophus Lie

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