Remark 1

Tag 00N2, permanent. Cite it as [Bourbaki, Tag 00N2].

Algebra, VIII, p. 390, § 20. Linear Representations of Algebras

Let $(\mathbf{V}, \pi)$ be a finite-dimensional linear representation of the algebra A. Let us introduce a basis $(e_1, \ldots, e_n)$ of V and the dual basis $(e_1^*, \ldots, e_n^*)$ of $\mathbf{V}^*$ . By the proof of Lemma 2, we have the relation

c(c_\pi(x, x^*)) = \sum_{k=1}^n c_\pi(e_k, x^*) \otimes c_\pi(x, e_k^*)

for $x\in V$ and $x^*\in V^*$ . For $1\leqslant i, j\leqslant n$ , set $\pi_{ij}=c_{\pi}(e_j, e^*_i)$ . For every $a\in A$ , the matrix of $\pi (a)$ with respect to the basis $(e_1, . . . , e_n)$ of V is equal to $(\pi_{ij}(a))$ . For $1\leqslant i\leqslant n$ and $1\leqslant j\leqslant n$ , we then have

c(\pi_{ij}) =\sum_{k=1}^n\pi_{ik}\otimes \pi_{kj} \tag{20}

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

Lemma 2