Derivations of important algebras? After knowing the importance of studying derivations of an algebras(Why we wonder to know all derivations of an algebra?), this problem naturally raised "what is the space of all derivations of important algebras like algebra of matrices, cliford algebra, exterior algebra and C* algebras?"
has anybody studied on this problem? In graded cases we have the notion of super and graded derivations.
does this kinds of derivations for famous algebras obtained?
If so, I want to be familiar with references with geometrical looking.
any hint is appreciated.
 A: There are many classes of algebras for which derivations have been studied.
Generally, if $A$ is an algebra and $a\in A$, the map $\delta_a:x\in A\mapsto ax-xa\in A$ is a derivation, which we call the inner derivation corresponding to $a$, and a somewhat boring one — for its existence does not tell us anything about the algebra. So usually what we do is consider the vector space $\def\Der{\operatorname{Der}}\Der(A)$ of all derivations, and its subspace $\def\InnDer{\operatorname{InnDer}}\InnDer(A)=\{\delta_a:a\in A\}$ and consider the quotient $\Der(A)/\InnDer(A)$ which, in a sense, tells us how many interesting derivations there are. We call $\Der(A)/\InnDer(AA)$ the vector space of outer derivations (even though its elements are not really derivations but classes of derivations) and write it $\def\OutDer{\operatorname{OutDer}}\OutDer(A)$.
The vector space $\OutDer(A)$ is an important invariant of $A$, and in fact it coincides with the so called first Hochschild cohomology group $HH^1(A)$ of $A$, which shows up all over the place.
If you want examples:


*

*If $A=M_n(k)$ is the algebra of $n\times n$ matrices over the ground field, then $HH^1(A)=0$. This means, precisey, that every derivation of $A$ is an inner derivation.

*If $L$ is a field extension of $k$ which is separable, then $HH^1(L)=0$ again. If the extension is nott separable, then the resul depends very much on what extension it is exactly.

*If $A$ is the Weyl algebra, that is, the algebra of differential operators (on the line, say) witth polynomial coefficients, then $HH^1(A)=0$. This is a famous result of Dixmier.

*If $A$ is an exterior algebra $\Lambda V$ on a vector space $V$, then on can easily compute $HH^1(A)$ but I do not recall the result.

*And so on and on. 
(My examples maybe make it seem like $HH^1$ is always zero, but it is certainly not!)
