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.

A Frobenius algebra whose bilinear form is symmetric is also called a symmetric algebra, but is not discussed here.

Construction

It turns out that S(V) is in effect the same as the polynomial ring, over K, in indeterminates that are basis vectors for V. Therefore this construction only brings something extra when the "naturality" of looking at polynomials this way has some advantage.

It is possible to use the tensor algebra T(V) to describe the symmetric algebra S(V). In fact we pass from the tensor algebra to the symmetric algebra by forcing it to be commutative; if elements of V commute, then tensors in them must, so that we construct the symmetric algebra from the tensor algebra by taking the quotient algebra of T(V) by the ideal generated by all differences of products