E0051 definition

Symmetric algebra

(1) Definition (Tensor algebra).

  • The tensor algebra over a vector space \(V\) is \(T(V) = \bigoplus_{k \geq 0} V^{\otimes k}\), with concatenation product and unit \(1 \in V^{\otimes 0} = \IR\).

(2) Definition (Symmetric algebra).

  • The symmetric algebra is the quotient \(\SS(V) = T(V) / (v \otimes w - w \otimes v)\). It is a commutative, associative, graded \(\IR\)-algebra with unit \(1 \in \SS^0(V) = \IR\).
  • \(\SS^k(V)\) is the \(k\)-th graded component (the image of \(V^{\otimes k}\) under the projection).
  • \(\SS_+(V) = \bigoplus_{k \geq 1} \SS^k(V)\) is the positive ideal.
  • \(\SH(V) = \prod_{k \geq 0} \SS^k(V)\) is the degree completion E0050.
  • Monomials are written \(v_1 \cdots v_k\) for the symmetric product of vectors \(v_i \in V\).

(3) Proposition (Universal property). For any linear map \(f: V \to A\) into a unital commutative algebra \(A\), there is a unique algebra morphism \(f^+: \SS(V) \to A\) extending \(f\).

Proof. \(T(V)\) has the universal property for associative algebras: \(f\) extends uniquely to an algebra map \(T(V) \to A\). Since \(A\) is commutative, this map kills the commutator ideal \((v \otimes w - w \otimes v)\) and descends to \(\SS(V)\).

(4) Proposition (Basis representation). If \((e_i)_{i \in I}\) is a basis of \(V\), then \(\SS^k(V)\) has basis \(\set{e^\nu : \nu \in \Map_f(I, \IN_0),\, |\nu| = k}\) with \(e^\nu = \prod_i e_i^{\nu(i)}\).

Proof. The monomials \(e^\nu\) span \(\SS^k(V)\) by commutativity. They are linearly independent because the symmetric product of basis vectors in \(V^{\otimes k}\) projects to distinct classes in \(\SS^k(V)\) (the quotient identifies only permutations of tensor factors).

(5) Proposition (Filtrations and direct sums).

  • The upper filtration \(F^K \SS(V) = \bigoplus_{k \geq K} \SS^k(V)\) is stable under multiplication: \(\mu(F^p \otimes F^q) \subseteq F^{p+q}\).
  • \(\SS(V \oplus W) = \SS(V) \otimes \SS(W)\).
  • \(\SH(V \oplus W) = \SH(V) \hat{\otimes} \SH(W)\).

Proof. Filtration stability: a product of homogeneous elements of degrees \(p\) and \(q\) has degree \(p + q\). Direct sums: by the universal property, the inclusions \(V \hookrightarrow V \oplus W\) and \(W \hookrightarrow V \oplus W\) induce an algebra map \(\SS(V) \otimes \SS(W) \to \SS(V \oplus W)\); it is an isomorphism because both sides have the same monomial basis in a basis of \(V \oplus W\). The completion statement follows degreewise.

Used by: