High Quality Content by WIKIPEDIA articles! 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 K-algebra 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. 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.

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