Lie Algebras from Associative Algebras
There is a great source for generating many Lie algebras: associative algebras. Specifically, if we have an associative algebra 
![\displaystyle[a,b]=ab-ba](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle%5Ba%2Cb%5D%3Dab-ba&bg=e6e6e6&fg=333333&s=0 "\displaystyle[a,b]=ab-ba")
In fact, this is such a common way of coming up with Lie algebras that many people think of the bracket as a commutator by definition.
Clearly this is bilinear and antisymmetric, but does it satisfy the Jacobi identity? Well, let’s take three algebra elements and form the double bracket
![\displaystyle\begin{aligned}\left[a,[b,c]\right]&=[a,bc-cb]\\&=a(bc-cb)-(bc-cb)a\\&=abc-acb-bca+cba\end{aligned}](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle%5Cbegin%7Baligned%7D%5Cleft%5Ba%2C%5Bb%2Cc%5D%5Cright%5D%26%3D%5Ba%2Cbc-cb%5D%5C%5C%26%3Da%28bc-cb%29-%28bc-cb%29a%5C%5C%26%3Dabc-acb-bca%2Bcba%5Cend%7Baligned%7D&bg=e6e6e6&fg=333333&s=0 "\displaystyle\begin{aligned}\left[a,[b,c]\right]&=[a,bc-cb]\\&=a(bc-cb)-(bc-cb)a\\&=abc-acb-bca+cba\end{aligned}")
We can find the other orders just as easily
![\displaystyle\begin{aligned}\left[a,[b,c]\right]&=abc-acb-bca+cba\\\left[c,[a,b]\right]&=cab-cba-abc+bac\\\left[b,[c,a]\right]&=bca-bac-cab+acb\end{aligned}](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle%5Cbegin%7Baligned%7D%5Cleft%5Ba%2C%5Bb%2Cc%5D%5Cright%5D%26%3Dabc-acb-bca%2Bcba%5C%5C%5Cleft%5Bc%2C%5Ba%2Cb%5D%5Cright%5D%26%3Dcab-cba-abc%2Bbac%5C%5C%5Cleft%5Bb%2C%5Bc%2Ca%5D%5Cright%5D%26%3Dbca-bac-cab%2Bacb%5Cend%7Baligned%7D&bg=e6e6e6&fg=333333&s=0 "\displaystyle\begin{aligned}\left[a,[b,c]\right]&=abc-acb-bca+cba\\\left[c,[a,b]\right]&=cab-cba-abc+bac\\\left[b,[c,a]\right]&=bca-bac-cab+acb\end{aligned}")
and when we add these all up each term cancels against another term, leaving zero. Thus the commutator in an associative algebra does indeed act as a bracket.
RSS Feeds
