Symmetric algebra

A generalization of a polynomial algebra. If $M$ is a unital module over a commutative associative ring $A$ with an identity, then the symmetric algebra of $M$ is the algebra $S (M) = T (M) / I$ , where $T (M)$ is the tensor algebra of $M$ and $I$ is the ideal generated by the elements of the form $x \otimes y - y \otimes x$ ( $x, y \in M$ ). A symmetric algebra is a commutative associative $A$ -algebra with an identity. It is graded: $S (M) = ⨁_{p \geq 0} S^{p} (M)$ where $S^{p} (M) = T^{p} (M) / (T^{p} (M) \cap I)$ , and $S^{0} (M) = A$ , $S^{1} (M) = M$ . The module $S^{p} (M)$ is called the $p$ -th symmetric power of the module $M$ . If $M$ is a free module with finite basis $x_{1}, \dots, x_{n}$ , then the correspondence $x_{i} \mapsto X_{i}$ ( $i = 1, \dots, n$ ) extends to an isomorphism of $S (M)$ onto the polynomial algebra $A [X_{1}, \dots, X_{n}]$ (see Ring of polynomials).

For any homomorphism $f : M \to N$ of $A$ -modules, the $p$ -th tensor power $T^{p} (f)$ induces a homomorphism $S^{p} (f) : S^{p} (M) \to S^{p} (N)$ (the $p$ -th symmetric power of the homomorphism $f$ ). A homomorphism $S (f) : S (M) \to S (N)$ of $A$ -algebras is obtained. The correspondences $f \mapsto S^{p} (f)$ and $f \mapsto S (f)$ are, respectively, covariant functors from the category of $A$ -modules into itself and into the category of $A$ -algebras. For any two $A$ -modules $M$ and $N$ there is a natural isomorphism $S (M \oplus N) = S (M) \otimes_{A} S (N)$ . If $M$ is a vector space over a field of characteristic 0, then the symmetrization $σ : T (M) \to T (M)$ defines an isomorphism from the symmetric algebra $S (M)$ onto the algebra $\tilde{S} (M) \subset T (M)$ of symmetric contravariant tensors over $M$ relative to symmetric multiplication: $x \lor y = σ (x \otimes y), x \in {\tilde{S}}^{p} (M), y \in {\tilde{S}}^{q} (M) .$

References[edit]

[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[edit]

The functor $S$ from $A$ -modules to commutative unitary $A$ -algebras solves the following universal problem. Let $M$ be an $A$ -module and $B$ a commutative unitary $A$ -algebra. For each homomorphism $f : M \to B$ of $A$ -modules there is a unique homomorphism $g : S (M) \to B$ of $A$ -algebras such that $g$ restricted to $S^{1} (M)$ coincides with $f$ . Thus, $S$ is a left-adjoint functor of the underlying functor from the category of commutative unitary $A$ -algebras to the category of $A$ -modules.