# what is the difference between algebra and Lie algebra

I am writting a simple repport on Lie algebras and I thought I could start just with a ''strandard'' simple definition of algebra to make an introduction for Lie algebra and then relate these 2 definitions somehow with one another.

Does anyone know how I can do it in a descriptive way?

Thank you

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## 1 Answer

Well, certainly a Lie algebra is not in general an algebra, since Lie brackets are not associative in general. The simplest relationship between associative algebras and Lie algebras is that given any algebra $A$ over a field $k$, defining $[x, y] = xy - yx$ for $x, y \in A$ turns $A$ into a Lie algebra.

But the traditional motivation for studying Lie algebras has less (directly) to do with associative algebras and more to do with Lie groups, although one can motivate the definition of a Lie algebra without formally introducing Lie groups by talking about derivations: I do so in this blog post. The perspective I take there is that Lie algebras are a natural way to talk about infinitesimal symmetry, just as groups are a natural way to talk about symmetry in the ordinary sense.

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thank you;) this explanaition makes sense –  Jacek May 7 '11 at 11:16
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