Proposition 8

Tag 00N1, permanent. Cite it as [Bourbaki, Tag 00N1].

Algebra, VIII, p. 388, § 20. Linear Representations of Algebras

On the vector space Θ(A), there exists a unique cogebra structure (III, §11, No. 1, p. 574) such that the mapping $\eta : A\rightarrow \Theta (A)^*$ is a homomorphism from A to the dual algebra (III, §11, No. 2, p. 579) of Θ(A). The cogebra Θ(A) is coassociative and admits $\varepsilon$ as a counit.

For every integer $n\geqslant 1$ , consider the K-linear mapping $j_n$ from $\Theta (A)^{\otimes n}$ to the dual of $A^{\otimes n}$ characterized by the formula

\langle j_n(f_1\otimes \cdots \otimes f_n), a_1\otimes \cdots \otimes a_n\rangle =\prod_{i=1}^n\langle f_i, a_i\rangle \tag{13}

for $(a_i)$ in $A^n$ and $(f_i)$ in $\Theta (A)^n$ . By Proposition 16, (ii) of II, §7, No. 7, p. 308, the mapping $j_n$ is injective. Denote by $m_K: K\otimes K\rightarrow K$ and $m_A: A\otimes A\rightarrow A$ the mappings deduced from the multiplication in K and A, respectively. For $f, g\in \Theta (A)$ and $a, b\in A$ , we have

\langle j_2(f\otimes g), a\otimes b\rangle =m_K\circ (\eta (a)\otimes \eta (b))(f\otimes g)

We therefore have

\langle j_2(t), a\otimes b\rangle =m_K\circ (\eta (a)\otimes \eta (b))(t) \tag{14}

for every $t\in \Theta (A)\otimes \Theta (A)$ .

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

Cited by

Nothing.

Cites

Proposition 16, (ii) of II, §7, No. 7, p. 308 not in this corpus
III, §11, No. 1, p. 574
III, §11, No. 2, p. 579