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.

About these ads

May 18, 2011 - Posted by | Algebra, Lie Algebras

6 Comments »

  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 [...]

    Pingback by The Lie Bracket of Vector Fields « The Unapologetic Mathematician | June 2, 2011 | Reply

  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 [...]

    Pingback by Lie Algebras Revisited « The Unapologetic Mathematician | August 6, 2012 | Reply

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

    Pingback by Linear Lie Algebras « The Unapologetic Mathematician | August 7, 2012 | Reply


Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

Follow

Get every new post delivered to your Inbox.

Join 392 other followers

%d bloggers like this: