Symmetric algebra of a vector space

For a vector space $V$, let $J$ be the two-sided ideal of $T(V)$ generated by

$v\otimes w-w\otimes v(v,w\in V)\mathrm{.}$

The symmetric algebra of $V$ is the commutative graded algebra

$S(V)=T(V)\mathrm{/}J\mathrm{.}$

The quotient is commutative because its degree-one generators commute, and all elements are sums of products of those generators. Since the relators are homogeneous of degree two, $J$ is homogeneous. Its degree-$n$ relation space is the span of adjacent-transposition differences: an arbitrary permutation difference is a telescoping sum along adjacent transpositions, and each adjacent difference is a degree-$n$ multiple of a generator of $J$. Thus the degree-$n$ image is exactly the quotient $S^n(V)$ from Symmetric and exterior powers over an arbitrary field. The degree-zero unit is the image of $1\in k$.