Difference between Free Lie algebra and universal enveloping algebra

Question

I have seen so many questions about free Lie algebra and universal enveloping algebra. As in this question (Universal envoloping algebra of a free Lie algebra.), a reader asked for the universal enveloping algebra of a free Lie algebra. I am following the book on Lie algebra written by Humphreys. In this book, he didn't give much about free Lie algebra and its construction. He said if $X$ is a set then to construct its free Lie algebra go through the vector space generated by X as a basis and consider this as a subalgebra of tensor algebra. I can not understand how to do this. Kindly explain this to me. There are so many questions on universal enveloping algebra on this site and I studied them but didn't get my answer.

Answer 1

First of all, note that an universal enveloping algebra is an associative algebra, wheras Lie algebras are seldom associative. So, it almost never happens that a universal enveloping algebra can also be a Lie algebra.

If $F$ is a set, the free Lie algebra on $F$ (over a field $k$) is basically a Lie algebra over $k$ which contains $F$ and which, besides that, satisfies the minimal possible amount of relations among its elements. So, for instance, if $F=\{X,Y\}$, then the free Lie algebra on $F$ contains $X$ and $Y$. It also contains $[X,Y]$ and $[Y,X]$ and, of course, $[X,Y]=-[Y,X]$, since this relation must hold for any Lie algebra. But it is not true that $[X,Y]=0$, since this doesn't hold in general for Lie algebras.