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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

\begin{displaymath}[f,g]=-[g,f],
\end{displaymath} (1)


\begin{displaymath}[f+g,h]=[f,h]+[g,h],
\end{displaymath} (2)

and
\begin{displaymath}[f,[g,h]]+[g,[h,f]]+[h,[f,g]]=0
\end{displaymath} (3)

(the Jacobi Identity), and are not Associative. The binary operation of a Lie algebra is the bracket
\begin{displaymath}[fg,h]=f[g,h]+g[f,h].
\end{displaymath} (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.




© 1996-9 Eric W. Weisstein
1999-05-25