Uncle of Witt algebra

Question

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.)

Answer 1

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 $[M_j,M_k]=M_k-M_j$ ($j,k\ne 0$).