The Unapologetic Mathematician

Mathematics for the interested outsider

Lie Algebras from Associative Algebras

There is a great source for generating many Lie algebras: associative algebras. Specifically, if we have an associative algebra A we can build a lie algebra L(A) on the same underlying vector space by letting the bracket be the “commutator” from A. That is, for any algebra elements a and b we define

\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}

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}

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.

May 18, 2011 Posted by | Algebra, Lie Algebras | 6 Comments

   

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