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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.

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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 $\mathfrak{S}(V) \cong K[x_1,\ldots,x_n]$.

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  • $\begingroup$ Johannes Hahn: Where does properties come from? I have used the book "Introduction to Lie algebras and Representation Theory-Humphreys", i didn't see those properties. $\endgroup$ Sep 1, 2019 at 21:06
  • $\begingroup$ I am confused. Can you help me to construct the elements from those two morphisms? $\endgroup$ Sep 1, 2019 at 21:32
  • $\begingroup$ As to "where do they come from": These are really well-known universal properties and immediately obvious to anyone who is familiar with the concept. You can find them where ever universal properties are discussed, say in introductory textbooks for category theory. But that is not necessary to solve the problem at hand. All you need to know is spelled out in my post. What do you mean by "construct the elements from those two morphisms" ? What elements? Do you mean the elements $a_1,\ldots,a_n$? Take a guess what they might be! It's really easy. (And then prove you guessed right of course) $\endgroup$ Sep 2, 2019 at 0:21
  • $\begingroup$ I mean the elements of $\mathfrak{G}(V)$ and $K[x1,\dots,xn]$ $\endgroup$ Sep 2, 2019 at 2:58
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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.

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    $\begingroup$ Despite the question's title and the fact that its subject, the symmetric algebra on $V$, can be viewed as a universal enveloping algebra, the question itself doesn't mention universal enveloping algebras (or Lie algebras) and seems to be considerably more elementary than the PBW theorem. $\endgroup$ Sep 1, 2019 at 18:53
  • $\begingroup$ Indeed, as Andreas says, the actual question is a lot more elementary, especially due to the fact that the defining ideal is generated purely in degree $2$, making the argument for injectivity easy. $\endgroup$ Sep 1, 2019 at 19:00

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