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Difference between revisions of "Symmetric algebra"

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A generalization of a polynomial algebra. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091590/s0915901.png" /> is a unital module (cf. [[Unitary module|Unitary module]]) over a commutative associative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091590/s0915902.png" /> with an identity, then the symmetric algebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091590/s0915903.png" /> is the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091590/s0915904.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091590/s0915905.png" /> is the [[Tensor algebra|tensor algebra]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091590/s0915906.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091590/s0915907.png" /> is the ideal generated by the elements of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091590/s0915908.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091590/s0915909.png" />). A symmetric algebra is a commutative associative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091590/s09159010.png" />-algebra with an identity. It is graded:
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A generalization of a polynomial algebra. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091590/s0915901.png" /> is a [[unital module]] over a commutative associative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091590/s0915902.png" /> with an identity, then the symmetric algebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091590/s0915903.png" /> is the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091590/s0915904.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091590/s0915905.png" /> is the [[Tensor algebra|tensor algebra]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091590/s0915906.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091590/s0915907.png" /> is the ideal generated by the elements of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091590/s0915908.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091590/s0915909.png" />). A symmetric algebra is a commutative associative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091590/s09159010.png" />-algebra with an identity. It is graded:
  
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091590/s09159011.png" /></td> </tr></table>
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091590/s09159011.png" /></td> </tr></table>

Revision as of 17:08, 11 April 2017

A generalization of a polynomial algebra. If is a unital 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:

References

[1] N. Bourbaki, "Eléments de mathématique" , 2. Algèbre , Hermann (1964) pp. Chapt. IV-VI
[2] A.I. Kostrikin, Yu.I. Manin, "Linear algebra and geometry" , Gordon & Breach (1989) (Translated from Russian)


Comments

The functor from -modules to commutative unitary -algebras solves the following universal problem. Let be an -module and a commutative unitary -algebra. For each homomorphism of -modules there is a unique homomorphism of -algebras such that restricted to coincides with . Thus, is a left-adjoint functor of the underlying functor from the category of commutative unitary -algebras to the category of -modules.

How to Cite This Entry:
Symmetric algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Symmetric_algebra&oldid=40936
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article