Symmetric algebra
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:
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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:
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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.
Symmetric algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Symmetric_algebra&oldid=40936