Understanding derivations on a Lie algebra

Question

I am very new to the subject of Lie algebras so excuse me if this is a rather stupid question.

I have some trouble understanding the Leibniz rule of a derivation $\delta$ on some Lie algebra $L$.

So a derivation $\delta$ is a linear map $\delta:L\rightarrow L$ that fullfills the Leibniz rule, i.e. $\delta(ab)=\delta(a)b+a\delta(b)$.

And here is my problem, how is the product $ab$ defined on a general vector space?

For $L=C(\mathbb{F})$ I know that $ab$ is just the product of two functions. But what do I do if $L=\mathbb{F}^n$? How to I define a product there?

Answer 1

You are new to Lie algebras, and not to arbitrary algebras with product $ab$. So here $ab$ is the Lie bracket $[a,b]$ and the Leibniz rules then reads as $$ D([a,b])=[D(a),b]+[a,D(b)] $$ for all $a,b\in L$. For the definition of a Lie algebra see here.

For a $K$-algebra $A$ the product on the vector space $A$ is given by the algebra, i.e., by the bilinear map $A\times A\rightarrow A$, $(a, b)\mapsto ab$.

Answer 2

On a Lie algebra $L$, the "multiplication" is the bracket operation $[\cdot,\cdot]$ that $L$ is equipped with, which must be bilinear, alternating, and satisfy the Jacobi identity (by definition). Usually, $L$ is a vector space of square matrices (or linear maps from a vector space to itself), equipped with the "commutator bracket" $[A,B]=AB-BA$.

Answer 3

The product $ab$ is not defined on an arbitrary vector space. If you have a bilinear map\begin{array}{ccc}L\times L&\longrightarrow&L\\(a,b)&\mapsto&ab\end{array}then a linear map $\delta$ from $L$ into itself is a derivation with respect to that bilinear map if$$(\forall a,b\in L):\delta(ab)=\delta(a)b+a\delta(b).$$