Symmetric Algebra

In mathematics, the symmetric algebra S(V) (also denoted Sym(V)) on a vector space V over a field K is the free commutative unital associative algebra over K containing V.

It corresponds to polynomials with indeterminates in V, without choosing coordinates. The dual, S(V*) corresponds to polynomials on V.

It should not be confused with symmetric tensors in V. A Frobenius algebra whose bilinear form is symmetric is also called a symmetric algebra, but is not discussed here.