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


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


We can find the other orders just as easily


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 […]

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  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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