Example

In particular the above can be applied when E is one of the graded cogebras T(M), S(M) or $\Lambda(M)$ for an A-module M (no. 1, Examples 5, 6 and 7). To find explicitly the bimodule structures thus obtained, we again identify a homogeneous element f of degree n in $T(M)^{*gr}$ (resp. $S(M)^{*gr}$, resp. $\Lambda(M)^{*gr}$) with an n-linear form (resp. symmetric n-linear form, resp. alternating n-linear form, also called an n-form) on $M^n$. It suffices to express the products

resp. $(x_1 \wedge x_2 \wedge \cdots \wedge x_p) \triangleleft f$ for every finite sequence $(x_i)_{1 \leq i \leq p}$ of elements of M and the analogues for the left inner product. Now, definitions (46) and (52) and formulae (3), (6) and (9) of no. 1 give respectively:

For $p \geq n$, we have respectively

(58) $(x_1 \otimes x_2 \otimes \cdots \otimes x_p) \triangleleft f = f(x_1, \ldots, x_n)x_{n+1} \otimes \cdots \otimes x_p$

(59) $(x_1 x_2 \ldots x_p) \triangleleft f = \sum_{\sigma} f(x_{\sigma(1)}, \ldots, x_{\sigma(n)})x_{\sigma(n+1)} \cdots x_{\sigma(p)}$

(60) $(x_1 \wedge x_2 \wedge \cdots \wedge x_p) \triangleleft f = \sum_{\sigma} \varepsilon_{\sigma} f(x_{\sigma(1)}, \ldots, x_{\sigma(n)})x_{\sigma(n+1)} \wedge \cdots \wedge x_{\sigma(p)}$

(where, in (59) and (60), the summations are taken over the permutations $\sigma \in \mathfrak{S}_p$ which are increasing on each of the intervals $[1, n]$ and $[n+1, p]$ of $\mathbf{N}$; and similarly

(61) $f \triangleright (x_1 \otimes x_2 \otimes \cdots \otimes x_p) = f(x_{p-n+1}, \ldots, x_p)x_1 \otimes x_2 \otimes \cdots \otimes x_{p-n}$

(62) $f \triangleright (x_1 x_2 \ldots x_p) = \sum_{\sigma} f(x_{\sigma(p-n+1)}, \ldots, x_{\sigma(p)})x_{\sigma(1)} \cdots x_{\sigma(p-n)}$

(63) $f \triangleright (x_1 \wedge x_2 \wedge \cdots \wedge x_p) = \sum_{\sigma} \varepsilon_{\sigma} f(x_{\sigma(p-n+1)}, \ldots, x_{\sigma(p)})x_{\sigma(1)} \wedge \cdots \wedge x_{\sigma(p-n)}$

(where, in (62) and (63), the summations are taken over the permutations $\sigma \in \mathfrak{S}_p$ which are increasing on each of the intervals $[1, p-n]$ and $[p-n+1, p]$ of $\mathbf{N}$).