Symmetric Algebra - Interpretation As Polynomials

Interpretation As Polynomials

Given a vector space V, the polynomials on this space are S(V*), the symmetric algebra of the dual space: a polynomial on a space evaluates vectors on the space, via the pairing .

For instance, given the plane with a basis {(1,0), (0,1)}, the (homogeneous) linear polynomials on K2 are generated by the coordinate functionals x and y. These coordinates are covectors: given a vector, they evaluate to their coordinate, for instance:

Given monomials of higher degree, these are elements of various symmetric powers, and a general polynomial is an element of the symmetric algebra. Without a choice of basis for the vector space, the same holds, but one has a polynomial algebra without choice of basis.

Conversely, the symmetric algebra of the vector space itself can be interpreted, not as polynomials on the vector space (since one cannot evaluate an element of the symmetric algebra of a vector space against a vector in that space: there is no pairing between S(V) and V), but polynomials in the vectors, such as v2 - vw + uv.