Universal enveloping algebras Let $V$ be a finite-dimensional vector space with the basis $\{x_1,\dots,x_n\}$, and let $\mathfrak{G}(V)=L(V)/I$ be an associative algebra, where $L(V)=\bigoplus_{i=0}^\infty V \bigotimes \cdots \bigotimes V (i \text{ copies})$ and $I$ is an ideal of $L(V)$ generated by all $x \otimes y-y \otimes x$, $(x,y \in V)$. My question is why $\mathfrak{G}(V)$ is canonically isomorphic to the polynomial algebra over $F$ in $n$ variables, with basis consisting of $1$ and all $x_{i(1)}, \ldots, x_{i(t)}$, $t \ge 1$, $1 \le i(1) \le \dots \le i(t) \le n$, $\{i: V \longrightarrow L(V) \text{ is an injection}\}$?
I have tried to construct a map $\varphi: \mathfrak{G}(V) \longrightarrow F[x_1,\dots,x_n]$, but it didn't work.
 A: What you have is the so called "symmetric algebra on $V$". It has the following universal property (which you should prove if it isn't immediately clear to you):

For all commutative $K$-algebras $A$ and all $K$-linear maps $f: V\to A$, there exists exactly one $K$-linear ring homomorphism $\hat{f}: \mathfrak{S}(V) \to A$ which satisfies $\hat{f}\circ i = f$.

The polynomial ring on the other hand has the following universal property (which you should also prove if it isn't already known to you):

For all commutative $K$-algebras $A$ and all elements $a_1,\ldots,a_n\in A$ there exists exactly one $K$-linear ring homomorphism $\phi: K[x_1,\ldots,x_n] \to A$ such that $\phi(x_i)=a_i$.

These properties yield morphisms $\mathfrak{S}(V) \to K[x_1,\ldots,x_n]$ and $K[x_1,\ldots,x_n]\to\mathfrak{S}(V)$ respectively. Now you can easily check that these two morphisms are inverse to each other so that they are in fact isomorphisms $\mathfrak{S}(V) \cong K[x_1,\ldots,x_n]$.
A: Well, this is the PBW basis theorem https://en.wikipedia.org/wiki/Poincaré–Birkhoff–Witt_theorem. 
It is quite hard to prove, but you can find a proof in the beginning of Dixmier's book Enveloping Algebras.
