Lie Algebra

A Nonassociative Algebra obeyed by objects such as the Lie Bracket and Poisson Bracket. Elements $f$, $g$, and $h$ of a Lie algebra satisfy

$$[f,g]=-[g,f],$$ (1)

$$[f+g,h]=[f,h]+[g,h],$$ (2)

and

$$[f,[g,h]]+[g,[h,f]]+[h,[f,g]]=0$$ (3)

(the Jacobi Identity), and are not Associative. The binary operation of a Lie algebra is the bracket

$$[fg,h]=f[g,h]+g[f,h].$$ (4)

See also Jacobi Identities, Lie Algebroid, Lie Bracket, Iwasawa's Theorem, Poisson Bracket

References

Lie Algebra

Jacobson, N. Lie Algebras. New York: Dover, 1979.