# The Unapologetic Mathematician

## 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

1. can you write about Engel’s Theorem

Comment by alan zhang | May 22, 2011 | Reply

2. I’m not actually covering Lie algebras in general yet; just enough to talk about vector fields.

Comment by John Armstrong | May 22, 2011 | Reply

3. […] happen if instead of using the regular composition product of these endomorphisms, we used the associated Lie bracket? We’d […]

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4. […] the vector space structures. Since is an associative algebra it automatically has a bracket: the commutator. Is this the same as the bracket on under this vector space isomorphism? Indeed it […]

Pingback by The Lie Algebra of a General Linear Group « The Unapologetic Mathematician | June 9, 2011 | Reply

5. […] I’ll finish the recap by reminding you that we can get Lie algebras from associative algebras; any associative algebra can be given a bracket defined […]

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6. […] algebra of endomorphisms — linear transformations from back to itself. We can use the usual method of defining a bracket as a […]

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