I am studying "Introduction to Lie algebra" written by "J.E. Humphreys". In chapter 10, when he is giving the concept of universal enveloping algebra, he introduces the notion of the symmetric algebra and says that it is isomorphic to the polynomial algebra. Please, someone, explain to me that what is polynomial algebra and how this symmetric algebra is isomorphic to polynomial algebra.
Where the symmetric algebra is defined in such a way if $V$ is vector space over some field $\mathbb{F}$ and $T(V)$ be its tensor algebra then $S(V)=T(V)/I$ is the symmetric algebra over V, where $I$ is the ideal generated by elements $(x\bigotimes y-y \bigotimes x).$
Best Answer
Let $x_1,...,x_n$ be a basis for $V$ (it may be infinite as well but for simpler notations I'll write it as finite); and let $f:V\to \mathbb F[X_1,...,X_n]$ be defined by $x_i\mapsto X_i$.
Then $f$ extends uniquely to an algebra morphism $T(V)\to \mathbb F[X_1,...,X_n]$. As $\mathbb F[X_1,...,X_n]$ is commutative, $I $ is in the kernel of this morphism and so it factors through $S(V)\to \mathbb F[X_1,...,X_n]$. It is moreover easy to show that $S(V)$ is commutative so that we may define $ \mathbb F[X_1,...,X_n]\to S(V), X_i\mapsto x_i$.
One then checks (on algebra generators) that these two maps are inverse to one another : $S(V)$ is the polynomial algebra on the $x_i$