Symmetric algebra

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A generalization of a polynomial algebra. If is a unital module (cf. Unitary module) over a commutative associative ring with an identity, then the symmetric algebra of is the algebra , where is the tensor algebra of and is the ideal generated by the elements of the form ( ). A symmetric algebra is a commutative associative -algebra with an identity. It is graded: where , and , . The module is called the -th symmetric power of the module . If is a free module with finite basis , then the correspondence ( ) extends to an isomorphism of onto the polynomial algebra (see Ring of polynomials).

For any homomorphism of -modules, the -th tensor power induces a homomorphism (the -th symmetric power of the homomorphism ). A homomorphism of -algebras is obtained. The correspondences and are, respectively, covariant functors from the category of -modules into itself and into the category of -algebras. For any two -modules and there is a natural isomorphism .

If is a vector space over a field of characteristic 0, then the symmetrization (cf. Symmetrization (of tensors)) defines an isomorphism from the symmetric algebra onto the algebra of symmetric contravariant tensors over relative to symmetric multiplication: How to Cite This Entry:
Symmetric algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Symmetric_algebra&oldid=18728
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article