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

### Other articles related to "symmetric algebra, algebra":

**Symmetric Algebra**- As A Universal Enveloping Algebra

... The

**symmetric algebra**S(V) is the universal enveloping

**algebra**of an abelian Lie

**algebra**, i.e ...

Glossary Of Tensor Theory - Algebraic Notation - Symmetric Power,

... This is the invariant way of constructing polynomial algebras. ...

**Symmetric Algebra**... This is the invariant way of constructing polynomial algebras. ...

### Famous quotes containing the word algebra:

“Poetry has become the higher *algebra* of metaphors.”

—José Ortega Y Gasset (1883–1955)

Main Site Subjects

Related Phrases

Related Words