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.
Read more about Symmetric Algebra: Construction, Interpretation As Polynomials, Categorical Properties, Analogy With Exterior Algebra, Module Analog, As A Universal Enveloping Algebra
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