A Witt algebra W is an infinite-dimensional Lie-algebra defined by the generator relations:  

W: $[L\_{j},L\_{k}]:=(j-k)\cdot L\_{j+k}$  

And my first thought was: What about the analogous algebra defined by  

W': $[L\_{j},L\_{k}]:=j\cdot{L\_k}-k\cdot{L\_j}$  

(Yes, I checked, the Jacobi relation is fulfilled.)  

There are about two possibilities: a) W' is uninteresting, question can be closed. b) W' is interesting and someone had the idea before me. Inventor's name suffices then. :-) (Unfortunately, there is no way to google such a formula.) 



Uncle of Witt algebra
Interesting/uninteresting is a very subjective thing, so let me try to just say several things that I see immediately. 


0) This algebra, unlike the Witt algebra, does not have any [obvious] grading, and this makes it in a sense less interesting for some purposes.


1) This algebra, unlike the Witt algebra, comes from antisymmetrising an *associative* product $L\_j\circ L\_k=j L\_k$ (in the case of the Witt algebra, it is not associative, but only satisfied the so called [left] pre-Lie identity $(a\circ b)\circ c-a\circ(b\circ c)=(b\circ a)\circ c-b\circ(a\circ c)$. So in a sense it is more unusual: Lie algebras do not frequently arise that way. 


2) The associative product above is not only associative but it does in fact satisfy the so called [left] permutative identity $a\circ b\circ c=b\circ a\circ c$ (so this algebra is very close to commutative; it is only non-commutative because there is no unit element). The operad of permutative algebras is the Koszul dual of the operad of pre-Lie algebras, so it is kind of an amusing coincidence. 


3) This Lie algebra contains a one-dimensional ideal spanned by $L\_0$, and the quotient by this ideal, after a linear change of basis $M\_j=\frac{L\_j}{j}$, has simpler commutation relations $[M\_j,M\_k]=M\_k-M\_j$ ($j,k\ne 0$). 


