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.

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 the differences of products

for all v and w in V.

In effect, S(V) is the same as the polynomial ring over K in indeterminates that are a basis for V.

Just as with a polynomial ring, there is a direct sum decomposition of S(V) as a graded algebra, into summands

which consist of the linear span of the monomials in vectors of V of degree k, for k = 0, 1, 2, ... (with S⁰(V) = K and S¹(V) = V). The K-vector space S^k(V) is the k-th symmetric power of V. (The case k = 2, for example, is the symmetric square and denoted Sym²(V).) It has a universal property with respect to symmetric multilinear operators defined on V^k.

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