Return to Question

I'm just studying Lie algebras. If $A$ is a $k$-algebra (not necessarily Lie or associative, just a bilinear law), it is straightforward to check that any derivation algebra of $A$ is a Lie algebra. I suppose that the converse is not true, but I can't find a counterexample.

Is there any example of a Lie algebra which is not isomorphic to the derivation algebra of any $k$-algebra (again, not necessarily associative or unital)?

I'm just studying Lie algebras. If $A$ is a $k$-algebra (not necessarily Lie or associative, just a bilinear law), it is straightforward to check that any derivation algebra of $A$ is a Lie algebra. I suppose that the converse is not true, but I can't find a counterexample.

1. Is there any example of a Lie algebra which is not isomorphic to the derivation algebra of any $k$-algebra (again, not necessarily associative or unital)?

2. Is there any example of a Lie algebra which is not isomorphic to the derivation algebra of any Lie algebra?

For the the second question the user YCor gives a positive answer. However, I am more interested in the first (and actually my original) question. Thank you!

I'm just studying Lie algebras. I have seen that Lie algebras over some field $k$ are quite often defined as derivation algebras of some $k$-algebra $A$ (not necessarily Lie or associative, just a bilinear law). It is straightforward to check that any derivation algebra is a Lie algebra. I suppose that the converse is not true, but I can't find a counterexample. Is there any example of a Lie algebra which is not isomorphic to the derivation algebra of any $k$-algebra?

I'm just studying Lie algebras. If $A$ is a $k$-algebra (not necessarily Lie or associative, just a bilinear law), it is straightforward to check that any derivation algebra of $A$ is a Lie algebra. I suppose that the converse is not true, but I can't find a counterexample. 

Is there any example of a Lie algebra which is not isomorphic to the derivation algebra of any $k$-algebra (again, not necessarily associative or unital)?

I'm just studying Lie algebras. I have seen that Lie algebras over some field $k$ are quite often defined as derivation algebras of some $k$-algebra $A$ (not necessarily Lie or associative, just a bilinear law). It is straightforward to check that any derivation algebra is a Lie algebra. I suppose that the converse is not true, but I can't find a counterexample. Is there any example of a Lie algebra which is not isomorphic to the derivation algebra of any Lie algebra?

I'm just studying Lie algebras. I have seen that Lie algebras over some field $k$ are quite often defined as derivation algebras of some $k$-algebra $A$ (not necessarily Lie or associative, just a bilinear law). It is straightforward to check that any derivation algebra is a Lie algebra. I suppose that the converse is not true, but I can't find a counterexample. Is there any example of a Lie algebra which is not isomorphic to the derivation algebra of any $k$-algebra?