Symmetric algebra

Remarks

$S(V)$ is a commutative associative unital $F$-algebra, graded by degree; its degree $k$ part $S^k(V)$ is the $k$-th symmetric power of $V$. Universal property: every linear map from $V$ to a commutative associative unital $F$-algebra $A$ extends uniquely to a unital algebra homomorphism $S(V) \to A$. Example: if $V$ has basis $x_1, \ldots, x_n$, then $S(V) \cong F[x_1, \ldots, x_n]$, and $S^k(V)$ has dimension $\binom{n + k - 1}{k}$; for $n = 2$, $k = 2$ the basis is $x_1^2, x_1 x_2, x_2^2$. Non-example: for $\dim V \ge 2$, the tensor algebra $T(V)$ itself is not commutative, so it is not $S(V)$. The exterior algebra uses instead the ideal generated by the elements $v \otimes v$.