How can I describe more concretely the free lie algebra over the singleton $X=\{x\}$? Is there any intuition on how to visualize the free Lie algebra when $X$ is more arbitrary?
By the definition I just know that the free Lie algebra over X is a Lie algebra $L(X)$ generated by $X$ which satisfies the universal property: for every map $\phi:X\rightarrow M, M$ Lie algebra, there exists a unique morphism of Lie algebras $\tilde \phi: L(X) \rightarrow M$ extending $\phi$.
This definition does not give me any intuition on how to describe $L(X)$ concretly . So i've tried to get some intuition via its construction. Well, $L(X)$ is just the subalgebra of the tensor algebra of $V=$ (vector space having $X$ as a basis) generated by $V$. But again for me it is not clear, especially when $X$ is a huge set.
I also have tried just to think about about a Lie algebra generated by $X$. Since $X$ is just a singleton, any Lie structure over the vector space $V$ must be abelian. And now? How does the property of being free over $X$ works here?
I really would like to get some intuition about this object. Thank you.
Best Answer
For $X=\{x\}$, $L(X)$ is the $1$-dim vector space spanned by $x$ with the structure of an abelian Lie algebra (well, any $1$-dim Lie algebra is abelian anyhow). It satisfies the universal property because any map $\phi:X\to M$, where $M$ is a Lie algebra, if $\tilde{\phi}:L(X) \to M$ is a Lie algebra homomorphism extending $\phi$, then $\tilde{\phi}$ is completely determined by its value at $x$, which is $\phi(x)$. So, $\tilde{\phi}$ is unique.
I am not sure what kind of intuitions you are looking for. But the section about Lyndon basis in the wikipedia page may be useful.